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WORKS  OF  PROF.  C.  E.  GREENE 


PUBLISHED   BV 


JOHN   WILEY   &  SONS. 


Graphics  for  Engineers,  Architects,  and  Builders. 

A   Manual   for   Designers,   and    a    Text-book  for 
Scientific  Schools. 

Trusses  and  Arches.  Analyzed  and  Discussed  by 
Graphical  Methods.     In  Three  Parts. 

Part  I.  Roof  Trusses.  Diagrams  for  Steady  Load, 
Snow,  and  Wind.     8vo,  cloth,  $1.25. 

Part  II.  Bridge  Trusses.  Single,  Continuous, 
and  Draw  Spans;  Single  and  Multiple  Systems; 
Straight  and  Inclined  Chords.     8vo,  cloth,  $2.50. 

Part  III.  Arches  in 'Wood,  Iron,  and  Stone.  For 
Roofs,  Bridges,  and  Wall  Openings  ;  Arched  Ribs 
and  Braced  Arches  ;  Stresses  from  Wind  and 
Change  of  Temperature.    8vo,  cloth,  $2.50. 

Published  by  the  A  uthor,  at  Ann  A  rbor.,  Mich, 

Structural  Mechanics:  The  Action  of  Materials 
Under  Stress.  A  work  on  the  Strength  and  Resist- 
ance of  Materials  and  the  Elements  of  Structural 
Design.  Ann  Arbor,  Mich.,  1897.  Printed  for  the 
author.    8vo,  300  pp.,  100  illustrations.    Price  $3.00. 


0|rc^^-'*-^ 


®rapl)ics  for  (Engineers,  3rcl)itcct0,  anb  i3uUbcr3: 

^  MANUAL  FOR  DESIGNERS,  AND  A   TEXT-BOUK  FOR    TECHNICAL  SCHOOLS 


TRUSSES  AND  AECHES 

ANALYZED    AND    DISCUSSED    BY  GEAPBICAL    METHODS 


CHARLES  E.  GREENE,  A.M.,  C.E., 

PROFBSSOR  OF  CIVIL  ENGINEERING,  UNIVERSITY  OF  MICHIGAN  ;    CONSULTING  ENQIMEEB. 


IN  THREE    PARTS. 

I. 

ROOF-TRUSSES  :   Diagrams  for  Steady  Load,  Snow,  and  Wind. 

iL 
BRIDGE-TRUSSES :    Single,  Continuous,  and    Draw  Spans  ;    Sinoi^k 
AND  Multiple  Systems;   Straight  and  Inclined  Chords. 

III. 
ARCHES,  IN  "Wood,  Iron,  and  Stone,  for  Roofs,  Bridges,  and  Wall- 
Openings  ;  Arched  Ribs  and  Braced  Arches  ;  Stresses  from  Wind 
AND  Change  of  Temperature;   Stiffened  Suspension  Bridges. 


Pakt  I.— roof-trusses. 

FOUB   FOLDING   PLATES. 


REVISED    EDITION. 
FIFTH   THOUSAND. 


NEW  YORK: 

JOHN   WILEY   &   SONS. 

London:  CHAPMAN  &  HALL,  Limited. 

1903. 


Engineering 

Ubvary 

COPTRISHT,    1890, 

By  CHARLES  E.  GREENE. 


En{nneeiii|g 

V.  I 


PREFACE  TO  PART  I. 


The  use  of  Graphical  Analysis  for  the  solution  of  problems 
in  construction  has  become  of  late  years  very  wide-spread. 
The  representation  to  the  eye  of  the  forces  which  exist  in  the 
several  parts  of  a  frame  possesses  many  advantages  over  their 
determination  by  calculation.  The  accuracy  of  the  figures  is 
readily  tested  by  numerous  checks.  Any  designer  who  fairly 
tries  the  method  will  be  pleased  with  the  simplicity  and 
directness  of  the  analysis,  even  for  frames  of  apparently  com- 
plex forms.  Those  persons  who  prefer  ariihmetical  computa- 
tion will  find  a  diagram  a  useful  check  on  their  calculations. 
Being  founded  on  principles  absolutely  correct,  these  diagrams 
give  results  depending  for  their  accuracy  on  the  exactness 
with  which  the  lines  have  been  drawn,  and  on  the  scale  by 
which  they  are  to  be  measured.  With  ordinary  care  the  dif- 
ferent forces  may  be  obtained  much  more  accurately  than  the 
several  parts  of  the  frame  can  be  proportioned. 

It  is  advisable  to  draw  the  figure  of  the  frame  to  quite  a 
large  scale,  as  the  lines  of  the  stress  diagram  are  drawn  paral- 
lel to  the  several  pieces  of  the  frame.  If  it  is  objected  by  any 
that  a  slight  deviation  from  the  exact  directions  will  materially 
change  the  lengths  of  some  of  the  lines,  and  therefore  give 
erroneous  results,  it  may  be  suggested  that  just  so  much 
change  in  the  form  of  the  frame  will  produce  this  change  in 
the  forces ;  one  is  therefore  warned  where  due  allowance  for 

V 


VI  PKEFACE  TO   PART   I. 

such  deformation  should  be  made  by  the  proper  distribution 
of  material.  The  comparison  of  different  types  of  truss  for 
the  same  locality  can  be  made  with  ease,  and  the  changes  pro- 
duced in  all  of  the  forces  in  any  frame  by  a  modification  of  a 
few  of  its  pieces  can  be  readily  shown.  By  applying  each  new 
principle  to  a  new  form  of  truss,  quite  a  variety  of  patterns 
have  been  treated  without  an  undue  multiplication  of  figures. 
The  method  of  notation  used  was  introduced  by  Mr.  Bow, 
in  his  "Economics  of  Construction."  The  diagrams,  as  here 
developed,  are  credited  in  England  to  Prof.  Clerk-Maxwell, 
and  the  method  is  known  by  his  name.  The  arrangement  of 
the  subjects,  the  application  of  the  method,  and  the  minor 
details  have  been  carefully  studied  by  the  author.  A  very 
limited  knowledge  of  Mechanics  will  enable  the  reader  to 
■snderstand  the  method  of  treatment  here  carried  out. 


NOTE  TO   KEVISED  EDITION. 


The  reception  of  this  Part  at  tlie  hands  of  teachers  and 
designers,  since  its  first  appearance  as  a  reprint  of  a  series  of 
articles  in  "  Eiigineering  News,"  has  been  so  hearty  and  sus- 
tained, that  it  has  been  thought  best  to  put  Egof-Tkusses  in  a 
uniform  dress  and  agreement  with  Bridge-Trusses  and  Arches. 
The  opportunity  has  been  seized  to  arrange  the  material  in  a 
more  systematic  order,  introduce  some  additional  problems, 
and  improve,  as  it  is  thought,  in  some  matters  of  detail. 

Quite  a  modification  has  been  made  in  the  way  of  regarding 
trusses  which  exert  horizontal  thrust,  and  Chapter  YIIL, 
Special  Solutions,  is  new.  The  solution  by  reversal  of  a  diag- 
onal has  been  used  in  the  author's  class-room  for  several  years. 
The  concluding  example  of  this  chapter  will  afford  a  good  test 
of  the  reader's  mastery  of  the  preceding  principles. 

C.  E.  G. 

Am?  Akbor,  Mich..  March  11, 1890. 


vu 


TABLE  OF  CONTENTS. 


CHAPTER  I. 

PAGES 

General  Principles.     Triangle  of  Forces  ;  Notation  ;  Illustrations,   .       1-6 

CHAPTER  II. 

Trusses  '^ith  Straight  Rafters  ;  Vertical  Forces.     Triangular,  King- 
post, and  Fink  Trusses, 7-15 

CHAPTER  III. 
Trusses  for  Flat  Roofs.     Queen-post,  "Warren,  and  Howe  Trusses,    .  16-21 

CHAPTER   IV. 

"Wind  Pressure  on  Pitched  or  Gable  Roofs.    Formula  for  Wind-pres- 
sure;  Examples  with  Roller  Bearings;  Wind  on  Alt  ernite  Sides,  23-32 

[CHAPTER  V. 

"Wind  Pressure  on  Curb  (or  Mansard)  and  Curved  Roofs.     Examples 

with  and  without  Rollers, 33-43 

CHAPTER  VI. 

Trusses  with  Horizontal  Thrust.    Scissor  and  Hammer-beam  Trusses,  44-49 

CHAPTER  VII. 
Forces  not  Applied  at  Joints, 50-52 

CHAPTER   VIII. 
Special  Solutions.    Reversal  of  Diagonal;  Trial  and  Error;  Example,  53-58 

viii 


TABLE   OF   CONTENTS.  ix 

CHAPTER   IX. 

Bending  Moment  and  Moment  of  Resistance.    Equilibrium  Polygon ; 
Graphical  Solution  for  Moment  of  Resistance,       ....  59-71 

CHAPTER  X. 

Load  and  Details.     "Weight  of  Materials ;  Allowable  Stresses  ;  Ties, 

,    Struts,  Beams,  Details, 72-77 


ROOF-TRUSSES. 


CHAPTER  I. 

GENERAL  PRINCIPLES. 

1.  Aim  of  the  Book. — It  is  proposed,  in  this  volume,  to 
explain  and  illustrate  a  simple  method  for  finding  the  stresses 
in  all  of  the  pieces  of  such  roof  or  other  trusses,  under  the 
action  of  a  steady  load,  as  permit  of  an  exact  analysis ;  to 
show  how  the  wind  or  any  oblique  force  alters  the  amount  of 
the  stresses  arising  from  the  weight ;  to  add  a  device  for 
solving  some  systems  of  trussing  which  otherwise  appear 
insoluble  by  the  above  method ;  and  to  conclude  with  such 
an  explanation  of  bending  moments  and  moments  of  resist- 
ance as  will  make  this  part  reasonably  complete  for  roof 
designing. 

2.  Triangle  of  Forces. — Taking  it  for  granted  that,  if  two 
forces,  acting  at  a  common  point,  are  represented  in  length 
and  direction  by  the  two  adjacent  sides  of  a  parallelogram  c  a 
and  c  n,  Fig,  2,  their  resultant  will  be  equal  to  the  diagonal  c  b 
of  the  figure,  drawn  from  the  same  point, — it  follows  that  a 
force  equal  to  this  resultant,  and  acting  in  the  opposite  direc- 
tion, will  balance  the  first  two  forces.  Hence,  considering 
one-half  of  the  parallelogram,  we  have  the  well-known  propo- 
sition that,  if  three  forces  in  equilibrium  act  at  a  single  point, 
and  a  triangle  be  drawn  with  sides  parallel  to  the  three  forces, 
these  sides  will  be  proportional  in  length,  by  a  definite  scale, 
to  these  forces.     The  forces  will  also  be  found  to  act  in  order 


2  EOOF-TKIISSES. 

round  tlie  triangle,  and  must  necessarily  lie  in  one  plane.  If 
the  magnitude  of  one  force  is  known,  the  other  two  can  be 
readily  determined. 

For  example  : — Let  a  known  weight  be  suspended  from  the 
points  1  and  2,  Fig.  1,  by  the  cords  1-3,  3-2,  and  3-4.  Draw 
c  b  vertically  to  represent  the  weight  by  any  convenient  scale 
of  pounds  to  the  inch.  This  line  will  then  be  parallel  to,  and 
will  equal  the  tension  in  3-4.  Draw  c  a  parallel  to  1-3,  and 
6 a  parallel  to  S-2.  Then  will  the  sides  of  the  triangle  cha 
represent  the  forces  which  act  on  the  point  3,  and  they  will  be 
found  to  follow  one  another  round  the  triangle,  as  shown  by 
the  arrows. 

3.  Notation. — A  notation  will  now  be  introduced  which 
will  be  found  very  convenient  when  applied  to  trusses  and 
diagrams.  In  the  frame  diagram  write  a  capital  letter  in 
every  space  which  is  cut  oflf  from  the  rest  of  the  figure  by 
lines,  real  or  imaginary,  along  which  forces  act.  See  Fig.  2 
and  following  figures.  Thus  D  represents  the  space  within 
the  triangular  frame,  A  the  space  limited  by  the  external 
forces  acting  at  1  and  2,  B  the  space  between  the  line  to  2 
and  the  line  which  carries  the  weight.  Then  let  that  piece  of 
the  frame  or  that  force  which  lies  between  any  two  letters  be 
called  by  those  letters  ;  thus,  the  upper  bar  of  the  triangle  is 
AD,  the  right  hand  bar  is  B  D,  the  cord  to  the  point  1  is  A  C, 
that  to  the  weight,  or  the  weight  itself,  is  C  B,  etc.  In  the 
diagrams  drawn  to  determine  the  magnitude  and  kind  of  the 
several  forces  acting  upon  or  in  the  frames  the  corresponding 
small  letters  wall  be  used  ;  thus  c  b  will  be  the  vertical  line 
representing  the  force  in  C  B,  &  a  the  tension  of  the  cord  B  A, 
and  ac  the  pull  on  1. 

4.  External  Forces. — Eeturning  to  Fig.  1,  let  us  suppose 
that  a  rigid,  triangular  frame  is  made  fast  to  those  cords,  so 
that,  as  shown  by  Fig.  2,  the  cords  are  attached  to  the  vertices 
of  the  triangle,  while  their  directions  are  undisturbed.  It  is 
evident  that  the  same  stresses  still  exist  in  those  cords,  if 
the  frame  has  no  weight,  and  that  the  portion  of  the  cords 


ROOF-TRUSSES.  3 

within  the  triangle  may  be  cut  away  without  destroying  the 
equilibrium  of  this  combination.  Hence  we  see  that  the  equi- 
librium of  this  frame  is  assured,  if  the  directions  of  these 
cords,  or  forces  external  to  the  frame,  meet,  if  prolonged,  at  a 
common  point. 

The  external  forces  C  B,  B  A  and  A  C,  taken  in  the  order 
C  B  A,  or  passing  around  the  exterior  of  the  triangle  in  a  direc- 
tion contrary  to  the  movement  of  the  hands  of  a  watch,  give 
the  triangle  of  forces c ha,  in  which c  b  acting  in  a  known  direc- 
tion, i.e.  downwards,  determines  the  direction  of  &  a  and  a  c  in 
relation  to  their  points  of  application  to  the  frame,  since  for 
equilibrium,  by  §  2,  they  must  follow  one  another  in  order 
round  the  stress  triangle. 

5.  Stresses  in  the  Frame. — Consider  the  left-hand  ajDex 
of  the  triangle.  This  point  is  in  equilibrium  under  the  action 
of  three  forces,  viz.,  those  in  A  C,  C  D,  and  D  A,  which  we  read 
around  the  point  in  the  same  order  as  be/ore ;  we  found  the 
direction  and  magnitude  of  A  C  in  the  previous  section,  and 
the  inclinations  of  the  other  two  are  known.  The  three  forces 
at  this  joint  must  therefore  be  equal  to  the  three  sides  of  a 
stress  triangle,  as  before. 

Begin  with  A  C,  the  fully  known  force,  and  pass  from  a  to  c, 
because  that  is  the  direction  of  the  action  of  the  force  A  C  on 
the  joint  under  consideration.  Next,  from  c,  draw  c  d  parallel 
to  C  D,  prolonging  it  until  a  line  from  its  extremity  d,  parallel 
to  the  piece  D  A,  will  strike  or  close  on  a.  The  stress  c  cZ  is 
found  in  C  D,  and  the  stress  d  a  exists  in  D  A.  The  direction 
in  which  we  passed  around  acd,  that  is,  from  c  to  d,  and  then 
to  a,  shows  that  C  D  and  D  A  both  exert  tension  on  the  joint 
where  they  meet. 

Next  take  the  lowest  joint.  Kemembering  again  to  take  the 
three  forces  in  equilibrium  here  in  the  order  in  which  the 
external  forces  were  taken,  and  commencing  with  the  first 
known  one,  we  go,  in  the  stress  diagram,  from  d  to  c;  because, 
since  we  have  just  found  that  c  d  represents  the  pull  of  C  D  on 
the  left-hand  apex  of  the  frame,  d  c  must  be  the  equal  and  op- 


4  KOOF-TRUSSES. 

posite  pull  of  D  C  on  the  lowest  joint.  Next  comes  c  h,  along 
wliich  we  pass  doivn,  the  direction  in  which  the  weight  acts ; 
and  finally  we  draw  from  b,bd  parallel  to  the  piece  B  D.  This 
last  line  will  close  on  the  point  d,  if  the  construction  has  been 
carefully  made,  and  the  direction  in  which  we  pass  over  it, 
from  b  to  d,  shows  that  the  piece  B  D  exerts  tension  on  the 
lowest  joint.  If  the  reader  will  now  run  over  the  triangle 
dba,  which  must  belong  to  the  right-hand  joint,  he  will  see 
that  the  directions  just  given  are  again  complied  with. 

The  reader  can  invert  Fig.  2 ;  then  the  weight  will  press 
down  upon  the  upper  apex  of  the  triangle,  and  he  will  find, 
upon  drawing  the  stress  diagram,  that  the  three  external 
forces  are  thrusts,  and  that  compression  exists  in  each  piece 
of  the  frame. 

6.  Second  Illustration:  External  Forces. — In  order  to 
make  these  first  principles  more  plain  let  us  take  another 
case.  Suppose  a  triangular  frame.  Fig.  3,  to  rest  against  a 
wall  by  one  angle,  to  have  a  weight  of  known  amount  sus- 
pended from  the  outer  corner,  and  to  be  sustained  by  a  cord 
attached  to  the  third  angle  and  secured  to  a  point  2.  Since 
this  frame  is  at  rest  under  the  action  of  three  external  forces 
which  are  not  parallel,  their  lines  of  action  must,  by  §  2,  meet 
at  one  common  point ;  and  since  the  known  directions  of  two 
of  these  forces,  AC  and  CB,  will  meet  at  4,  if  prolonged,  the 
force  exerted  on  the  frame  by  the  wall  at  1  must  have  the 
direction  of  the  line  1-4.  The  magnitude  and  kind  of  the 
two  unknown  external  forces  therefore  will  be  found  by  the 
following  construction,  observing  the  rules  of  interpretation 
already  laid  down  : — 

Draw  ac,  vertically  down,  equal  to  the  known  weight  and 
force  A  C ;  next,  from  c,  a  line  parallel  to  the  cord  and  force 
CB,  and  prolong  it  until,  from  its  extremity  b,  a  line  may  be 
drawn  parallel  to  BA,  to  strike  a.  As  we  went  from  c  to  b, 
and  from  6  to  a,  C  B  must  pull  on,  and  B  A  must  thrust  against, 
the  frame. 

7.  Stresses  in  the  Frame. — Take  whichever  joint  is  most 


ROOF-TRUSSES.  5 

convenient,  for  instance  the  one  wliere  the  weight  is  attached  ; 
pass  down  ac  for  the  external  force  and  then,  obser\ing  the 
order  in  which  the  triangle  of  external  forces  was  drawn,  draw 
cd  parallel  to  C  D  and  da  parallel  to  DA.  Since  cd,  in  the 
triangle  acd  (made  up  of  forces  ac,cd,  and d a),  must  represent 
a  force  acting  upwards,  C  D  exerts  tension  on  this  joint ;  and, 
similarly,  d  a  (not  a  d)  shows  that  D  A  thrusts  against  the  same 
joint. 

Take  next  the  joint  at  1.  Here  the  reaction,  as  before  as- 
certained, is  h  a  ;  next  comes  a  d,  the  thrust  of  the  piece  A  D 
against  this  joint ;  and  lastly  d  h,  drawn  parallel  to  D  B,  to 
close  on  h  the  point  of  beginning,  shows  that  D  B  also  thrusts 
with  this  amount  at  1. 

8.  Third  Illustration. — Once  more,  suppose  that  the  tri- 
angular frame,  Fig.  4,  has  a  weight  attached  to  its  lowest 
angle  and  that  the  two  other  points  are  supj)orted  by  inclined 
posts.  The  forces  1-4  and  2-4  must  intersect  3-4  at  the  same 
point.  Draw  a  b  vertically  downwards,  and  equal  to  the  given 
weight ;  draw  b  c  jDarallel  to  2-4  or  B  C  and  c  a  parallel  to  1-4 
or  C  A.  Hence  be  and  ca  are  thrusts.  For  the  lowest  joint, 
after  passing  down  a  b  for  the  weight,  draw  b  d  parallel  to  B  D 
and  d  a  parallel  to  D  A,  thus  finding  that  B  D  and  D  A  both 
pull  on  the  joint  A  B,  and  hence  are  tension  members.  As  in 
former  cases,  find  d  c,  which  proves  to  be  compression. 

9.  General  Application. — Since,  in  Mechanics,  the  poly- 
gon of  forces  follows  naturally  from  the  triangle  of  forces, 
being  simply  a  combination  of  several  triangles,  the  same 
rules  will  apply  when  we  have  to  deal  with  several  external 
forces  or  a  number  of  pieces  meeting  at  one  joint.  1°.  Draw 
the  polygon  of  external  forces  for  the  whole  frame,  taking 
them  in  order  round  the  truss,  either  to  the  left  or  right,  as 
may  seem  convenient,  2°.  Take  any  joint  where  not  more 
than  two  stresses  in  the  pieces  are  unknown,  and  draw  the 
polygon  of  forces  for  it.  Treat  the  pieces  and  external  forces 
which  meet  at  the  joint  in  that  order,  to  the  left  or  right,  in 
which  the  external  forces  were  taken,  and  begin,  if  possible, 


6  ROOF-TRUSSES. 

witli  tlie  first  kno\vn  force,  so  that  tlie  two  unknown  forces  will 
be  the  last  two  sides  of  that  particular  pol^^gon.  3°.  The  di- 
rection in  which  any  line  is  passed  over,  in  going  round  the 
polygon  as  above  directed,  shows  whether  the  stress  in  the 
piece  to  which  that  line  was  drawn  parallel  acts  towards  or 
from  the  joint  to  which  the  polygon  belongs,  and  hence  is 
compression  or  tension.  The  reader  must  understand  this 
lDrinci]3le  in  order  to  correctly  interpret  his  diagrams. 

10.  Reciprocal  Figures. — Prof.  Clerk-Maxwell  called  the 
frame  and  stress  diagrams  reciprocal  figures  ;  for,  referring  to 
the  figures  already  drawn,  we  see  that  the  forces  which  meet 
at  one  point  in  the  frame  diagram  give  us  a  triangle  or  closed 
polygon  in  the  stress  diagram,  and  the  pieces  which  make  the 
triangular  frame  have  their  stresses  represented  by  the  lines 
which  meet  at  one  point  in  the  stress  diagram.  The  same 
reciprocity  will  exist  in  more  complex  figures,  and  it  is  one 
of  the  checks  which  we  have  upon  the  correctness  of  our 
diagrams. 

The  convenience  of  the  notation  explained  in  §  3  depends 
upon  the  above  property. 


CHAPTEK  II. 

TRUSSES  WITH  STRAIGHT  RAFTERS;  VERTICAL  FORCES. 

11.  Triangular  Truss;  Inclined  Reactions. — Suppose 
that  the  roof  represented  in  Fig.  5  has  a  certain  load  jjer  foot 
over  each  rafter,  and  let  the  whole  weight  be  denoted  by  W. 
It  is  evident  that  one-half  of  the  load  on  the  rafter  C  F  will 
be  supported  by  the  joint  B  and  one-half  by  the  ujDper  joint ; 
the  same  will  be  true  for  the  rafter  D  F ;  therefore  the  joint 
B  will  carr}'  ^  W,  the  upper  joint  ^  W,  and  the  joint  at  E  ^^  W. 
The  additional  stress  produced  in  C  F  by  the  bending  action 
of  the  load  which  it  carries  is  not  considered  at  this  time,  but 
must  be  noticed  and  allowed  for  separately.  (See  Chap.  IX.) 
Taking  the  external  forces  in  order  from  right  to  left  over  the 
roof,  lay  off  ed,  or  |-W,  vertically,  to  represent  the  weight 
E  D  acting  downward  at  the  joint  E,  nest  d  c  equal  to  |-  W, 
for  the  weight  D  C,  and  lastly  c  b  for  the  weight  at  B.  Call 
eb  the  load  line. 

Let  the  two  reactions  or  supporting  forces  for  the  present 
be  considered  as  a  little  inclined  from  the  vertical,  as  shown 
by  the  arrows  B  A  and  A  E.  Since  the  truss  is  symmetrical 
and  symmetrically  loaded,  the  resultant  of  the  load  must  j)ass 
through  the  apex  of  the  roof,  and,  as  the  two  supporting  forces 
must  meet  this  resultant  at  one  point,  the  two  reactions  must 
be  equally  inclined.  Then,  to  complete  the  polygon  of  ex- 
ternal forces : — as  we  have  drawn  ed,  dc,  and  c 6  in  order, 
passing  over  the  frame  to  the  left, — draw  next  b  a,  ujd  from  the 
extremity  b  of  the  load  line,  and  parallel  to  the  upward  reac- 
tion B  A ;  and  lastly  a  line  a  e,  parallel  to  the  other  reaction 
A  E,  to  close  on  e,  the  point  of  beginning. 

12.  Triangular  Truss :  Stresses. — While  in  this  truss  we 
might  find  the  stresses  at  any  joint,  let  us  begin  at  B.     Here 

7 


8  ROOF-TEUSSES. 

we  have  equilibrium  under  the  action  of  four  forces,  of  which 
the  two  external  ones  are  known.  Taking  the  latter  in  the 
same  order  as  above,  and  beginning  at  c  (§  9,  2°),  pass  over 
ch  downwards  and  ha  upwards;  then  draw  af  parallel  to 
AF,  in  such  a  direction  that/c,  drawn  from  /parallel  to  F  C, 
will  strike  c,  the  point  of  beginning.  Because  we  passed  from 
a  to/,  AF  will  pull  on  the  joint  B,  and  as  we  then  passed 
from/  to  c,  F  C  will  exert  a  thrust  on  B.  (It  is  usual  to  draw 
a/ from  a  and  /c  from  c  till  they  meet  at/;  but  to  determine 
the  kind  of  stress,  one  must  pass  over  the  lines  in  the  direc- 
tions noted.) 

Passing  next  to  the  apex  of  the  roof,  and  again  taking  the 
forces  in  the  same  order,  pass  down  the  line  dc  for  the  ex- 
ternal force,  thence  up  to  /  for  the  thrust  c/  and  finally 
draw  fd  parallel  to  F  D,  thus  determining  the  thrust  of  that 
rafter  against  the  top  joint.  If  this  line  does  not  close  on  d, 
the  drawing  has  not  been  made  with  care.  As  all  the  stresses 
are  now  found  we  need  not  examine  the  remaining  joint.  It 
may  again  be  noted  that  we  pass  over  a  stress  line  in  one 
direction  when  we  analyze  the  stresses  at  the  joint  at  one  end 
of  the  piece  to  which  the  line  is  parallel,  and  in  the  reverse 
direction  when  we  consider  the  joint  at  the  other  end  of  the 
same  j^iece. 

13.  Effect  of  Inclined  Reactions. — If  the  supporting 
forces  had  been  more  inclined  from  the  vertical,  the  point  a, 
of  their  meeting  in  the  stress  diagram,  would  have  been  nearer 
/,  thus  diminishing  the  tension  in  A  F,  but  not  affecting  the 
compression  in  the  rafters.  The  inclination  might  be  so  much 
increased  that  o,  would  fall  on  /  when  the  piece  A  F  would 
have  no  stress,  the  thrust  of  the  rafters  being  balanced  with- 
out it.     If  a  fell  to  the  right  of  /  af  would  be  a  thrust. 

14.  Triangular  Truss :  Vertical  Reactions.— If  the  two 
reactions  are  vertical,  as  will  be  the  case  when  the  roof  truss 
is  simply  placed  upon  the  wall,  B  A  and  A  E,  Fig.  6,  will  each 
be  -1^  W,  and  the  point  a  will  therefore  be  found  at  the  middle 
of  e  h.     The  polygon  of  external  forces  has  closed  up  and  be- 


ROOF-TRUSSES.  9 

come  a  straight  line,  but  in  the  analysis  it  must  still  be  used. 
Thus  we  pass  down  ed-\-dc-^cb  for  the  weights  at  the  joints 
and  back  over  ba-\-ae  for  the  reactions.  The  rest  of  the 
diagram  follows  from  §  12. 

The  diagrams  which  the  reader  draws  may  be  inked  in  black 
and  red,  one  denoting  compression,  the  other  tension,  or  the 
two  kinds  of  stress  may  be  indicated  by  the  signs  -j-  and  — . 

15.  King-post  Truss. — In  the  truss  of  Fig.  7  the  rafters  are 
supported  at  points  midway  between  their  extremities.  Each 
point  of  junction  of  two  or  more  pieces  is  considered  a  joint 
around  which  the  pieces  would  be  free  to  turn  were  they  not 
restrained  by  their  connections  with  other  points.  Whatever 
stiffness  the  joint  may  possess  from  friction  between  its  parts, 
or  from  the  continuity  of  a  piece,  such  as  a  rafter,  through  the 
joint,  is  not  taken  into  account,  and  may  add  somewhat  to  the 
strength  of  the  truss. 

In  this  example,  therefore,  half  of  the  uniform  load  on  C  L 
will  be  carried  at  B,  and  be  represented  by  the  arrow  B  C  ;  the 
other  half  together  with  half  of  the  uniform  load  on  D  K  will 
make  the  force  C  D,  and  so  on,  three  of  the  joints  carrying 
each  one-quarter  of  the  whole  load,  and  the  two  extreme  ones 
one-eighth  each. 

On  a  vertical  line  lay  off  gf=^Vl,fe  =  ed  =  dc  =  ^W 
and  c  &  =  |-  W  ;  then  ba  =  ag  =  ^'W,  the  two  supporting  forces. 
In  the  order  shown  by  the  arrow,  for  the  joint  B  we  have  c  b 
external  load,  b  a  reaction ;  then  draw  a  I,  tension,  §  9,  3°,  par- 
allel to  AL  and  Ic,  compression,  parallel  to  LC.  At  the 
joint  C  D  the  unknown  forces  now  are  those  in  L  K  and  K  D. 
Begin  with  the  load  dc,  following  with  cl,  the  stress  just 
found  in  C  L ;  then  draw  I  Jc,  compression,  parallel  to  L  K, 
and  k  d,  compression,  parallel  to  K  D,  to  close  on  d.  Passing 
next  to  the  joint  D  E,  ed  is  the  load,  d  k  the  thrust  of  D  K  on 
this  joint,  k  i  the  tension  in  K  I,*  and  i  e,  to  close  on  e,  is  the 
compression  in  I E.     Take  next  the  joint  in  the  middle  of  the 

*  It  will  be  seen  that  K I  is  a  tension  member  or  tie,  and  not  a  post  as  would 
be  inferred  from  the  name  given  to  this  truss  by  old  builders. 


10  EOOF-TRTJSSES. 

lower  tie ;  we  know  i  h,  k  I,  and  I  a ;  the  next  stress  lies  in 
AH;  as  we  have  just  arrived  at  a  from  I,  we  must  pass  back 
horizontally  until  a  line  from  h  parallel  to  H I  \d\\  close  on  i,  the 
point  from  which  we  started.  The  remaining  line  lif  is  easily- 
determined  by  taking  either  the  joint  E  F  or  the  one  at  G. 

It  will  be  noticed  that,  since  the  truss  is  symmetrically 
made  and  loaded,  the  stress  diagram  is  symmetrical ;  k  i  must 
be  bisected  by  « ? ;  dk  and  e  i  must  intersect  on  a  I.  Atten- 
tion to  such  points  ensures  the  accuracy  of  the  drawing. 

A  truss,  Fig.  8,  is  now  submitted,  which  the  reader  is  advised 
to  analyze  for  himself,  as  a  test  whether  the  principles  thus  far 
explained  are  clearly  understood. 

16.  Wooden  Truss  with  Frequent  Joints. — The  truss 
represented  by  Fig.  9,  a  simple  extension  of  Fig.  7,  is  one  well 
adapted  for  construction  in  timber,  the  verticals  alone  being 
made  of  iron.  It  can  be  used  for  roofs  of  large  span.  In  any 
actual  case,  before  beginning  to  draw  the  diagram,  assume  an 
approximate  value  for  the  weight  of  the  truss,  add  so  much  of 
the  weight  of  the  purlins,  small  rafters,  boards  and  slates,  or 
other  covering,  as  is  supported  by  one  truss,  and  divide  this 
total  weight  by  the  number  of  equal  parts,  such  as  D  I  or  E  L, 
in  the  two  rafters.  We  thus  obtain  the  weight  which  is  sup- 
posed to  act  at  each  joint  where  two  pieces  of  the  rafter  meet. 
The  weight  at  each  abutment  joint  will  be  half  as  much.  If 
the  rafter  is  not  supported  at  equidistant  points,  divide  the 
total  load  by  the  combined  length  of  both  rafters,  to  obtain  the 
load  per  foot  of  rafter,  and  then  multiply  the  load  per  foot  by 
the  distance  from  the  middle  of  one  piece  of  the  rafter  to  the 
middle  of  the  next,  to  obtain  the  load  on  the  joint  which 
connects  them.  Numerical  values  will  be  introduced  in  later 
chapters. 

Draw  the  vertical  load  line  equal  to  the  total  weight,  and 
beginning  with  6  c  as  the  load  on  B  from  one-half  of  C  H, 
space  off  the  weights  cd,  de,  etc.,  in  succession,  closing  at  p 
with  a  half  load  as  at  b.  The  point  of  di^dsion  «,  at  the 
middle  of  p  b,  marks  off  the  two  supporting  forces  p  a  and  a  &, 


EOOF-TRUSSES.  11 

which  close  the  polygon  of  external  forces.  Beginning  now 
at  B,  draw,  as  heretofore  directed,  §  9,  abcha  for  this  joint. 
The  order  of  these  letters  gives  the  directions  of  the  forces  on 
the  joint  B.  Then  for  the  joint  C  D  we  have  h  c  d  i  h ;  for  H  K 
we  have  a  h  i  k  a  ;  for  D  E  we  have  k  i  d  e  I  k,  etc.  Observe 
that,  by  taking  the  joints  in  this  order,  first  the  one  on  the 
rafter,  and  then  the  one  below  it  on  the  tie,  we  have  in  each 
case  only  two  unknown  forces,  out  of,  at  some  joints,  five 
forces.  We  repeat,  also,  the  remark  that  it  is  expedient,  when 
possible,  first  to  pass  over  all  the  known  forces  at  any  joint, 
taking  them  in  the  order  observed  with  the  external  forces 
when  laying  off  the  load  line.  The  rest  of  the  diagram  pre- 
sents no  difficulty. 

After  the  stress  in  N  O  is  obtained,  the  diagram  will  begin 
to  repeat  itself  inversely,  the  stress  in  O  G  being  equal  to  that 
in  F  N.  It  is  therefore  unnecessary  to  draw  more  than  one- 
half  of  this  figure,  except  for  a  check  on  the  accuracy  of  the 
drawing  by  the  intersections  which  are  seen  on  inspection  of 
this  diagram.  Noting  the  stresses  found  in  the  several  poly- 
gons, we  see  that  all  the  inclined  pieces  are  in  compression, 
while  the  horizontal  and  vertical  members  are  in  tension. 

17.  Superfluous  Pieces. — Sometimes  a  vertical  rod  is  in- 
troduced in  the  first  and  last  triangles,  where  dotted  lines  are 
drawn.  It  is  e%'ident  that  this  rod  will  be  of  no  service  if  all 
the  load  is  assumed  to  be  concentrated  on  the  joints  of  the 
rafters,  and  this  fact  can  be  determined  from  the  stress  dia- 
gram as  well.  Thus,  taking  the  joint  below  H,  Fig.  9,  we 
have  three  forces  in  equilibrium ;  begin  at  a  in  the  stress  dia- 
gram and  pass  to  h  along  the  line  already  found  for  A  H  ;  then 
we  are  required  to  draw  a  vertical  line  from  h  and,  from  its 
extremity,  a  horizontal  line  to  close  on  the  point  a  from  which 
we  started ;  the  vertical  line  therefore  can  have  no  length. 
All  that  this  vertical  rod  can  do  is  to  keep  the  horizontal  tie 
from  sagging,  by  sustaining  whatever  small  weight  is  found 
at  its  foot. 

Therefore,  whenever  there  are  at  a  joint  but  three  pieces  or 


12  ROOF-TRUSSES. 

lines  along  wliich  forces  can  act,  and  two  of  these  pieces  lie 
in  one  straight  line,  it  follows  from  the  above  that  the  third 
piece  must  be  without  stress,  and  that  the  first  two  pieces  or 
lines  will  have  the  same  stress.  Thus,  L  K  of  Fig.  7  and 
H I  of  Fig.  9  would  have  no  compression  if  the  external  load 
C  D  were  removed.  This  fact  will  often  prove  of  ser^ace  in 
analysis. 

18.  Problem. — Draw  the  stress  diagram  for  the  truss  illus- 
trated by  Fig.  10,  which  is  supported  on  a  shoulder  at  the 
wall  and  by  an  overhead  tie  running  from  the  right  end.  It 
will  be  convenient  to  imagine  that  tie  replaced  by  the  inclined 
reaction  shown  by  the  arrow  at  the  right,  as  thus  the  reaction 
is  kept  on  the  right  of  the  load  at  that  joint.  The  reaction  at 
the  wall  will  cut  the  tie  where  the  resultant  of  the  load  cuts 
it ;  if  the  load  is  uniform  over  the  rafter,  that  intersection  is 
at  the  middle  of  the  tie. 

Next,  try  this  problem  with  the  two  inclined  diagonals 
reversed,  so  as  to  slant  up  to  the  right.  Notice  the  upper 
left-hand  joint.  Compare  the  two  cases,  as  to  difference  in 
magnitude  and  kind  of  stress. 

19.  Joints  where  three  Forces  are  Unknown. — It  ap- 
pears impracticable  to  determine  the  stresses  at  any  joint  where 
more  than  two  forces  are  unknown.  In  Fig.  9,  we  could  not 
start  with  the  joint  C  D  or  at  D  E  ;  for  we  should  know  only 
the  external  force  or  load,  and  have  three  unknown  stresses  to 
find ;  therefore  our  quadrilateral,  of  wliich  one  side  is  known, 
might  have  the  other  sides  of  various  lengths,  but  still  parallel 
to  the  original  pieces  of  the  frame.  When  the  joints  were 
taken  in  the  order  observed  this  difficulty  was  not  met  with. 

When,  in  some  cases,  we  find  three  or  more  apparently  un- 
known forces  at  a  joint  we  may  have  some  knowledge  of  the 
proportion  which  exists  between  one  or  more  of  them  and  a 
known  force,  and  can  thus  determine  the  proper  length  of  the 
line  in  the  stress  diagram.  An  example  of  such  a  case  will  be 
given  in  Fig.  11.     In  Chapter  YIII.  will  be  found  a  treatment 


KOOF-TRUSSES.  13 

that  is  applicable  to  certain  trusses  which  otherwise  offer  diffi- 
culties in  solution. 

20.  Polonceau  or  Fink  Truss. — Fig.  11  shows  a  truss 
which  is  often  built  in  iron.  The  loads  at  the  several  joints 
of  the  rafters  are  found  by  the  method  prescribed  in  §  16. 
It  will  be  unnecessary  to  dwell  ujjon  the  manner  of  finding 
the  stresses  at  the  joints  B,  C  D,  and  H  K,  for  which  the 
stresses  will  be  ch,  It  a,  nk,  ki,  hi  and  id.  But  when  we 
attempt  to  analyze  the  joint  D  E,  we  find  that,  with  the  ex- 
ternal load,  we  have  six  forces  in  equilibrium,  of  which  those 
along  E  M,  M  L,  and  L  K  are  unknown.  If  we  try  the  joint 
L  A  we  find  four  forces,  three  of  which  are  at  present  unknown. 
We  are  therefore  obliged  to  seek  some  other  way  of  determin- 
ing one  of  the  stresses. 

It  will  be  seen,  upon  inspection,  that  the  joint  E  F  is  like 
the  joint  C  D  ;  and  it  will  appear  reasonable  that  X  M  should 
have  an  equal  stress  with  I H.  We  may  then  expect  that 
there  must  be  as  much  and  the  same  kind  of  stress  exerted  by 
M  L  to  keep  the  foot  of  the  strut  N  M  from  moving  laterally 
as  is  found  necessary  in  K I  to  restrain  the  foot  of  I H. 

Returning  then  to  the  joint  D  E,  and  beginning  with  k  iy 
pass  next  over  i  d,  then  d  e,  then  draw  e  m,  parallel  to  E  M,  to 
such  a  point  m,  that  (ha^-ing  drawn  m  I  until  its  extremity  I 
comes  in  the  middle  of  what  will  be  the  space  between  e  m 
and  fn,  or  until  m  I  equals  in  leng-th  i  k),  the  line  I  k  shall  close 
on  k  whence  we  started.  The  ties  and  struts  can  be  readily 
selected  by  the  direction  of  movement  over  these  lines  in 
reference  to  the  joint  D  E.  The  remaining  joints  when  taken 
in  the  usual  order  of  succession  offer  no  difiiculty,  and  the 
other  half  of  the  diagram  need  not  be  added,  unless  one  de- 
sires a  check  on  the  results. 

This  truss  will  be  treated  again  in  §  7-4. 

The  polygon  which  we  have  just  traced,  kidemlk,  affords 
a  good  illustration  of  the  rule  that  the  forces  which  meet  at  a 
joint  make  a  closed  polygon  in  the  stress  diagram.  The  sym- 
metry of  the  triangles  hik  and  mnl,  and  their  resemblance  to 


14  EOOF-TRUSSES. 

k  1 0,  are  wortli  noting,  and  will  assist  one  in  drawing  diagrams 
for  trusses  of  this  type. 

21.  Cambering  the  Lower  Tie. — Sometimes  it  is  tliought 
desirable  to  raise  the  tie  A  O,  either  to  give  more  height  be- 
low the  truss  or  to  improve  its  appearance.  The  effect  on  the 
stresses  of  such  an  alteration  is  very  readily  traced,  and  one 
then  can  judge  how  much  change  it  is  exj)edieut  to  make. 
Let  it  be  proposed  to  raise  the  portion  A  O  of  the  tie  to  the 
position  indicated  by  the  dotted  line,  and  thus  to  introduce 
such  changes  in  the  other  members  that  they  shall  coincide 
with  the  other  dotted  lines  in  Fig.  11,  while  the  load  remains 
unchanged. 

The  line  c  li  for  joint  B  now  becomes  cli',  being  prolonged 
until  h'  a  can  be  drawn  parallel  to  H  A  in  its  new  position. 
Next  come  h' i'  and  i' d;  then  we  easily  draw  i'k',  h'V,  I'm', 
m'  n',  etc.  The  struts  H  I,  K  L,  and  M  N  are  the  only  pieces 
in  this  half  of  the  truss  unaffected  by  the  change  ;  the  amount 
of  increase,  and  the  serious  increase,  of  the  other  stresses  for 
any  considerable  elevation  of  the  lower  member  can  be  readily 
seen. 

22.  Load  on  all  Joints. — If  one  prefers  to  consider  that 
a  portion  of  the  weight  of  the  truss,  or  that  a  floor,  ceiling 
or  other  load  is  supported  at  the  lower  joints,  the  load 
may  be  distributed  as  in  Fig.  12.  Here  the  joints  Q  R  and 
R  S  carry  their  share  of  the  weight  of  the  pieces  which  touch 
these  joints,  as  well  as  such  other  load  as  may  properly  be 
put  there.  Each  supporting  force,  if  the  load  is  symmetrical, 
will  still  be  one-half  the  total  load,  but  the  two  will  no  longer 
divide  the  load  line  equally,  nor  can  the  load  line  be  at  once 
measured  off  as  equal  to  the  total  weight. 

Begin,  if  convenient,  with  the  extremity  H  of  the  truss,  and 
lay  off  hi,  ik,  kl,  etc.,  downwards,  ending  with  op.  Passing 
on,  around  the  truss,  lay  off  next  the  reaction  p  q  upwards, 
equal  to  one-half  the  total  weight,  then  q  r  and  r  s  downwards, 
and  finally  s  h  upwards,  for  the  other  supporting  force,  to  close 
on  h.     The  polygon  of  external  forces,  therefore,  doubles  back 


EOOF-TRUSSES.  15 

on  itself  as  it  were,  and  lip  is  still  the  load  on  the  exterior  of 
the  roof.  The  diagram  can  now  be  drawn,  by  taking  three 
joints  on  the  rafter  in  succession  before  trying  the  joint  Q II ; 
when  taking  that  joint  remember  that  there  is  a  load  upon  it. 
The  loads  on  the  horizontal  tie  cause  the  stresses  in  its  three 
parts  to  be  drawn  as  three  separate  lines,  instead  of  being 
superimposed  as  in  the  figures  before  given. 

A  diagram  may  now  be  drawn  for  Fig.  13.  The  upper  part 
of  the  roof,  dotted  in  the  figure,  throws  its  load,  through  the 
small  rafters,  on  the  upper  joints  of  the  truss. 

23.  Stresses  by  Calculation — It  is  evident,  from  insi:)ection  of  the  pre- 
ceding diagrams,  that  the  stresses  may  be  calculated  by  means  of  the 
known  inclinations  of  the  parts  of  the  trusses.  The  degree  of  accuracy 
with  which  they  can  be  scaled  equals,  however,  if  it  does  not  exceed  the 
approximation  which  designing  and  actual  construction  make  to  the  theo- 
retical structure. 

24.  Distribution  of  Load  on  the  Joints. — In  Unwin's  "  Iron  Bridges  and 
Roofs"  the  rafter  is  treated  as  a  beam  continuous  over  three  or  more 
supports,  and  the  distribution  of  the  load  on  the  several  joints  is  there 
determined  by  that  hypothesis.  That  such  an  analysis  may  be  true,  it  is 
necessary  that  all  the  points  at  which  the  rafter  is  supported  shall  remain 
in  definite  positions,  usually  a  straight  line.  As  slight  deformations 
of  the  truss  and  unequal  loading  of  the  joints  will  prevent  the  realiza- 
tion of  that  assumption,  a  division  of  the  load  at  any  point  of  a  rafter  or 
other  piece  so  that  the  joints  at  its  two  ends  shall  be  loaded  in  the  inverse 
ratio  of  the  two  segments  into  which  the  point  divides  the  piece  will  best 
represent  the  case.  Uniform  loads  will  be  distributed  easily  by  §  16.  A 
different  distribution  of  the  load,  however,  if  one  prefers  it,  will  only  re- 
quire a  corresponding  division  of  the  load  line.  (See  Part  II.,  Bridge 
Trusses,  Chaps.  VIII.  and  IX.) 


w&fAKTMENT  OF  CIVIL  ENGINEE^"|MU 


CHAPTEK  III. 

TRUSSES  FOR  FLAT  ROOFS. 

25.  Trapezoidal  Truss;  Equal  Loads. — A  consideration 
of  tlie  trapezoidal,  or  queen-post,  truss,  rej^resented  by  Fig.  14, 
will  bring  out  two  or  three  points  whicb  will  be  of  use  in  the 
analysis  of  other  trusses.  In  this  case,  let  us  suppose  the 
load  to  be  on  the  lower  part,  or  bottom  chord,  of  the  truss. 
In  order  to  separate  the  supporting  forces  from  the  small 
weights  on  the  ends  of  the  truss,  and  to  permit  them  to  come 
consecutively  with  the  other  weights  in  the  load  line,  let  us 
draw  the  supporting  forces  above  the  tie,  instead  of  below  as 
before.  The  rectangle  formed  by  the  two  vertical  and  two 
horizontal  pieces  might  become  distorted ;  we  will  therefore 
introduce  the  brace  H  I,  represented  by  the  full  line.  The 
rectangle  is  thus  divided  into  two  triangles  and  movement  pre- 
vented. The  dotted  line  shows  a  piece  which  might  have  been 
introduced  in  place  of  the  other. 

If  the  truss  is  symmetrically  loaded,  or  C  D  =  D  E,  we  shall 
get  the  first  stress  diagram.  The  stress  in  each  vertical  is 
here  seen  to  be  the  load  at  its  foot.  The  stress  in  the  piece 
H  I  proves  to  be  zero.  If  the  load  had  been  on  the  upper 
joints,  no  stress  would  have  been  found  in  the  verticals  also. 
(See  §  17.)  It  is  evident  that  a  trapezoidal  truss,  when  sym- 
metrically loaded,  requires  no  interior  bracing.  This  fact 
might  readily  be  seen  if  we  considered  the  form  assumed  by  a 
cord,  suspended  from  two  points  on  a  level,  and  carrying  two 
equal  weights  symmetrically  placed. 

26.  Trapezoidal  Truss;  Unequal  Loads. — The  second 
stress  diagram  will  be  drawn  when  the  weight  C  D  is  less 
than   D  E.     Let  us  suppose  that  b  c  and  ef  are  of  the  same 

16 


EOOF-TRUSSES.  17 

magnitude  as  in  the  first  diagram,  and  let  the  span  of  the 
truss,  or  distance  between  supports,  which  we  shall  denote  by 
Z,  be  di\dded  by  the  joints  into  three  equal  parts.  The  first 
step  is  to  find  the  suj^porting  forces.  If  each  external  force 
be  multiplied  by  the  perpendicular  distance  of  its  line  of  ac- 
tion from  any  one  assumed  point,  which  distance  may  be  called 
its  leverage,  and  all  the  products  added  together,  those  which 
tend  to  produce  rotation  about  this  point  in  one  direction 
being  called  plus,  and  those  tending  the  other  way  minus,  it 
is  necessary  for  equilibrium  that  the  sum  of  these  products 
shall  be  zero  ;  otherwise  the  rotation  can  take  place.  A  con- 
venient point  to  which  to  measure  the  distances  will  be  one  of 
the  points  of  support,  for  instance  the  right-hand  one.  Then 
we  shall  have 

Ar.Z-rE.Z-ED.tZ-DC.iZ-CB.O  +  BA.O  =  0, 
or 

AF.Z  =  rE.Z  +  ED.tZ  +  DC.i?; 
therefore 

AF  =  FE  +  tED+iDO. 

If  E  D  be  taken  as  3  D  C, 

AF  =  FE  +  |ED. 

It  will  be  seen  that  the  object  in  taking  the  point  or  axis  at  B 
is  to  eliminate  B  A,  and  have  only  one  unknown  quantity,  A  F. 
This  method  of  determination  is  called  taking  moments,  and  is  at 
once  the  simplest  and  most  generally  aj)plicable.  Lay  off  the 
above  reaction  at  fa ;  a  b  will  be  the  reaction  at  the  right 
support.  One  cause  of  a  diagram's  failure  to  close,  when  drawn 
by  a  beginner,  is  carelessness  in  placing  the  reactions  on  the 
load  line  in  the  wrong  order. 

The  point  a  being  now  located,  we  can  proceed  to  draw  the 
second  diagram.  The  construction  requires  no  explanation ; 
but  we  will  call  attention  to  the  fact  that  a  compressive  stress 
here  exists  in  H  I.  If,  in  place  of  the  diagonal  represented 
by  the  full  line,  the  one  shown  by  the  dotted  line  is  now  sup- 
plied, the  reader  can  without  difficulty  trace  out  for  himself 


18  KOOF-TRUSSES. 

the  change  in  the  diagram,  which  is  denoted  by  the  dotted 
lines  and  the  letters  marked  by  accents.  The  stress  in  this 
diagonal  will  be  seen  to  be  tensile.  Changing  the  diagonal 
reverses  its  stress. 

It  is  also  worthy  of  notice  that  the  only  pieces  affected  by 
the  substitution  of  one  diagonal  for  the  other  are  those  which 
form  the  quadrilateral  enclosing  the  diagonals.  This  fact 
will  be  of  service  later. 

27.  Use  of  Two  Diagonals.— If,  at  another  time,  this  ex- 
cess of  load  might  fall  on  C  D  in  place  of  D  E,  the  stress  on 
either  diagonal  would  be  reversed  :  that  is,  if  it  sloped  down 
to  the  right  it  would  be  a  tie  ;  if  to  the  left,  a  strut.  As  a  ten- 
sion diagonal  is  likely  to  be  a  slender  iron  rod,  which  is  of  no 
practical  value  to  resist  a  thrust,  while  the  compression  mem- 
ber, unless  made  fast  at  its  extremities,  will  not  transmit  ten- 
sion, a  weight  or  force  which  may  be  shifted  from  one  joint  to 
another  may  require  the  designer  to  introduce  two  diagonals 
in  the  same  rectangle  or  trapezium,  or  else  to  so  proportion 
and  fasten  one  diagonal  as  to  withstand  either  kind  of  stress. 

Where  both  diagonals  occur  the  diagram  can  still  be  drawn. 
Determine  which  kind  of  stress,  tension  or  compression,  the 
two  shall  be  designed  to  resist,  and  then,  when  drawing  the 
diagram,  upon  arriving  at  a  particular  panel  or  quadrilateral, 
try  to  proceed  as  if  only  one  of  the  diagonals  existed.  If  a 
contrary  kind  of  stress  to  the  one  desired  is  found  to  be 
needed,  erase  the  lines  for  this  panel  only,  and  take  the  other 
diagonal.  In  the  treatment  for  wind  pressure,  this  method 
becomes  serviceable,  since  the  wind  may  blow  on  either  side  of 
the  roof. 

This  truss  can  be  used  for  a  bridge  of  short  span. 

28.  Trusses  for  Halls. — It  is  sometimes  the  case  that,  in 
covering  a  large  building,  it  is  desired  to  have  the  interior 
clear  from  columns  or  partitions,  while  a  roof  of  very  slight 
pitch  is  all  that  is  needed.  As  it  is  not  expedient  to  have  a 
truss  of  much  depth,  since  the  space  occupied  by  it  is  not 
generally  available  for  other  purposes,  one  of  several  types  of 


ROOF-TRUSSES.  19 

parallel-chord  bridge  trusses  may  be  employed,  for  instance 
the  "  Warren  Girder,"  of  Fig.  15,  which  is  an  assemblage  of 
isosceles  triangles.  In  a  j^ublic  hall,  galleries  may  be  sus- 
pended from  the  roof,  and  the  weight  of  a  heavy  panelled  or 
otherwise  ornamented  ceiling  may  be  added  to  what  the  truss 
is  ordinarily  expected  to  carry.  The  depth  ma}-  be  less  than 
here  drawn,  but,  for  clearness  of  figure,  we  have  not  made  the 
truss  shallow. 

If  the  roof  pitches  both  ways  from  the  middle  of  the  span, 
the  top  chord  may  conform  to  the  slope,  making  the  truss 
deeper  at  the  middle  than  at  the  ends ;  l)ut  a  light  frame  may 
be  placed  above,  as  shown  by  the  dotted  lines,  and  supported 
at  each  joint  of  the  top  chord.  The  straight-chord  truss  is 
more  easily  framed.  If  the  roof  pitches  slightly  transversely 
to  the  trusses,  it  will  be  convenient  to  make  them  all  of  the 
same  depth  and  put  on  some  upper  works  to  give  the  proper 
slope.  The  ends  of  the  truss  could  readily  be  adapted  to  a 
mansard  roof. 

29.  Warren  Girder. — In  Fig.  15,  each  top  joint  is  sup- 
posed to  be  loaded  with  the  weight  of  its  share  of  roof,  in 
which  case  the  joint  LM  or  PQ  will  have  three-quarters  of 
the  weight  on  N  O  or  O  P,  if  the  roof  is  carried  out  to  the 
eaves  as  marked  on  ihe  left ;  or  practically  the  same  as  N  O, 
if  the  roof  follows  the  line  I L.  The  bottom  joints  are  sup- 
posed to  carry  the  weight  of  the  ceiling,  and  in  addition  the 
tension  of  a  suspending  rod  to  a  gallery  on  each  side.  The 
load  line  will  be  equal  to  the  weight  on  the  upper  part  of  the 
truss,  and  the  polygon  of  external  forces  will  overlap,  as  in 
Fig.  12,  previously  exj^lained,  §  22.  We  go  from  k  to  r,  for 
the  loads  on  the  exterior  in  sequence,  then  up  to  s  for  the 
left-hand  reaction,  then  down  to  lo  for  the  loads  on  the 
interior,  and  finally  close  on  k  with  the  right-hand  reaction. 

Upon  drawing  the  diagram  it  -^dll  be  seen  that  the  stress  is 
compression  in  the  top  chord  and  tension  in  the  bottom  chord ; 
that  the  stresses  in  the  chords  increase  from  the  supports  to 
the  middle ;  that  the  stresses  in  the  braces  decrease  from  the 


20  KOOF-TKUSSES. 

ends  of  the  truss  to  the  middle,  and  that  alternate  ones  are  in 
compression  and  in  tension,  those  which  slant  up  from  the 
abutment  towards  the  centre  being  compressed,  and  those 
which  incline  in  the  other  direction  being  in  tension.  The 
tie-braces  are,  therefore,  A B,  C  D,  F  G,  and  HI.  A  decrease 
of  depth  in  the  truss  will  increase  the  stresses  iu  the  chords. 

30.  Howe  Truss;  Determination  of  Diagonals.  —  A 
truss  with  parallel  chords  may  be  employed,  in  which  the 
braces  are  alternately  vertical  and  inclined.  The  designer 
will  choose  whether  the  verticals  shall  be  ties  and  the  diag- 
onals struts,  in  which  case  the  type  is  called  the  "Howe 
Truss,"  Fig.  16,  or  the  verticals  struts  and  the  diagonals  ties, 
when  it  is  known  as  the  "Pratt  Truss."  There  is  an  advan- 
tage in  having  the  struts  as  short  as  possible,  but,  if  one 
desires  to  use  but  little  iron,  the  Howe  is  a  good  form. 

To  decide  which  diagonal  of  the  rectangle  shall  be  occupied 
by  the  piece  : — Start  from  the  wall  as  a  fixed  point ;  it  is  evi- 
dent that,  to  keep  the  load  C  D  from  sinking,  C  Q  must  be  a 
strut.  If  we  wish  to  put  a  tie  in  this  panel,  it  must  lie  in  the 
other  diagonal,  shown  by  the  dotted  line.  CD  now  being 
held  in  place,  P  O  as  a  strut  will  uphold  D  E.  We  thus  may 
work  out  from  each  wall  until  we  have  passed  as  much  load 
as  equals  the  amount  supported,  or  the  reaction,  at  that  wall. 
If  the  last  load  passed  exactly  comj)letes  the  amount  required 
to  equal  the  reaction,  no  diagonal  will  be  required  in  the  next 
panel.  We  might  draw  diagonals,  one  in  each  panel,  sloping 
in  either  direction  as  we  pleased,  and  then  construct  the  stress 
diagram.  If  we  found  a  stress  in  any  diagonal  opposite  to 
the  stress  we  desired,  §  27,  we  could  then  erase  that  diagonal 
and  substitute  the  other,  erasing  also  so  much  of  the  diagram 
as  referred  to  the  pieces  in  that  panel.  Were  the  chords  not 
parallel,  this  method  might  be  necessary  (see  Fig.  20),  but  in 
the  present  case  it  is  better  to  draw  the  load  line  fi.rst,  find  the 
dividing  point  ff,  Fig.  16,  for  the  two  reactions,  see  what  load 
it  cuts,  and  then  incline  the  diagonals  from  each  wall  either 
up  or  down,  as  preferred,  towards  that  loaded  joint. 


ROOF-TRUSSES.  21 

31.  Howe  Truss  ;  Diagram. — In  the  present  example  C  D 
is  supj)osed  to  be  four  times  D  E,  etc.  A  tower  on  that  end 
of  the  truss  or  some  suspended  load  will  account  for  the  dif- 
ference. Eecalling  the  manner  in  which  the  supporting  forces 
were  found  when  the  load  was  unsymmetrical,  §  2G,  use  a 
panel  as  a  unit  of  distance,  call  a  panel  length  p  and  the  ordi- 
nary weight  on  a  joint  lo.  Then  we  shall  have,  taking  moments 
about  H, 

w  .2)0-  +[2  +  3)  +  4  ?^  .  4^?  +  I-  li; .  0J5  =  R .  o^^,     or     R  =  4.9  w, 

the  reaction  at  B,  or  a  h.  The  two  supporting  forces  will  then 
be  A  a  and  a  h.  Draw  the  stress  diagram  as  usual ;  the  di- 
agonals will  all  come  in  compression  as  intended,  and  the 
verticals  will  be  ties.  There  will  plainly  be  no  stress  in  the 
dotted  vertical  O  N.  The  stress  in  the  chords  is  inversely 
proj)ortional  to  the  depth  of  the  truss,  and  economy  of  ma- 
terial in  the  chords  will  be  served  by  making  the  depth  as 
much  as  j)ossible,  within  reasonable  limits.  In  bridge  trusses 
this  depth  is  seldom  less  than  from  one-sixth  to  one-eighth  of 
the  span. 

32.  Moving  Load. — If  the  joint  D  E  also  might  become 
hea^dly  loaded,  we  could  draw  another  diagram  for  that  case, 
and,  as  the  joints  in  succession  had  their  loads  increased,  we 
might  make  as  many  diagrams.  From  a  collection  of  dia- 
grams for  all  positions  of  a  mo^ang  load,  we  could  select  the 
maximum  stress  for  each  piece.  A  truss  designed  to  resist 
such  stresses  would  answer  for  a  bridge.  We  should  find  that 
the  greatest  stresses  in  the  chords  occurred  in  all  panels  when 
the  bridge  was  hea^dly  loaded  throughout,  and  that  the  great- 
est stress  in  a  diagonal  was  found  when  the  bridge  was  heaAdly 
loaded  from  this  piece  to  one  end  only,  that  end  generally 
being  the  more  distant  one.  As  we  have  more  expeditious 
methods  of  analyzing  a  bridge  truss,  this  one  is  not  used. 
The  graphical  treatment  of  bridge  trusses  is  found  in  Part  XL 
of  this  work. 


CHAPTEE  ly. 

WIND   PRESSUEE   ON   PITCHED   ROOFS. 

33.  Action  of  Wind. — The  forces  liitlierto  considered  have 
been  vertical ;  the  wind  jDressure  on  a  roof  is  inclined.  It  was 
once  usual  to  deal  with  the  pressure  of  the  wind  as  a  vertical 
load,  added  to  the  weight  of  the  roof,  snow,  etc.,  and  the 
stresses  were  obtained  for  the  aggregate  pressure.  This  treat- 
ment manifestly  cannot  be  correct.  The  wind  may  be  taken 
without  error  as  blowing  in  a  horizontal  direction  ;  it  exerts  its 
greatest  pressure  when  blowing  in  a  direction  at  right  angles 
to  the  side  of  a  building ;  it  consequently  acts  ujDon  but  one 
side  of  the  roof,  loads  the  truss  unsymmetrically,  and  some- 
times causes  stresses  of  an  opposite  kind,  in  parts  of  the 
frame,  from  those  due  to  the  steady  load.  Braces  which  are 
inactive  under  the  latter  weight  may  therefore  be  necessary 
to  resist  the  force  of  the  wind. 

It  will  not  be  right  to  design  the  roof  to  sustain  the  whole 
force  of  the  wind,  considered  as  horizontal ;  nor  wdll  it  be  cor- 
rect to  decompose  this  horizontal  force  into  two  rectangular 
components,  one  perpendicular  to  the  roof,  and  the  other 
along  its  surface,  and  then  take  the  perpendicular  or  normal 
comj)onent  as  the  one  to  be  considered ;  for  the  pressure  of 
the  wind  arises  from  tlie  imj)act  of  particles  of  air  moving 
■with  a  certain  velocity,  and  these  particles  are  not  arrested, 
but  only  de\dated  from  their  former  direction  upon  striking 
the  roof.  Yet  the  analysis  aj^plicable  to  a  jet  of  water  striking 
an  inclined  surface  cannot  be  used  here,  for  water  escapes 
laterally  against  the  air,  a  comparatively  unresisting  medium, 
while  the  wind  particles,  if  we  may  so  term  them,  deflected  by 
the  roof,  are  turned  off  against  a  stream  of  similar  air,  also  in 
motion,  which  retards  their  lateral  progress  and  thus  causes 

22 


EOOF-TRUSSES.  23 

them  to  press  more  strongly  against  the  roof.  We  are  obliged, 
therefore,  to  have  recourse  to  experiments  for  our  data,  and 
from  them  to  deduce  a  formula. 

34.  Formula  for  Wind  Pressure. — It  appears  that,  for  a 
given  pressure  exerted  by  a  horizontal  wind  current  on  any 
square  foot  of  a  vertical  plane,  the  pressure  against  a  plane 
inclined  to  its  direction  is  perj^endicular  to  the  inclined  sur- 
face, and  is  greater  than  the  normal  component  of  the  given 
horizontal  pressure.  Unwin  quotes  Hutton's  experiments  as 
showing  that,  if  P  equal  the  horizontal  force  of  the  wind  on  a 
square  foot  of  a  vertical  plane,  the  perpendicular  or  norma] 
pressure  on  a  square  foot  of  a  roof  surface  inclined  at  an 
angle  i  to  the  horizon  may  be  expressed  by  the  empirical 

formula 

P  sin  ^■l•**'=°s'■-^ 

If,  then,  the  maximum  force  of  the  wind  be  taken  as  40 
pounds  on  the  square  foot,  representing  a  velocity  of  from  80 
to  90  miles  per  hour,  the  normal  pressure  per  square  foot  on 
surfaces  inclined  at  different  angles  to  the  horizon  will  be  : 


Angle  of 
Roof. 

Normal 
Pressure. 

Angle  of 
Roof. 

Normal 
Pressure. 

5° 

5.2  lbs. 

35° 

30.1  lbs. 

10 

9.6 

40 

33.4 

15 

14.0 

45 

36.1 

20 

18.3 

50 

38.1 

25 

22.5 

55 

39.6 

30 

26.4 

■60 

40.0 

For  steeper  pitches  the  pressure  may  be  taken  as  40  pounds. 

Any  component  in  the  plane  of  the  roof,  from  the  friction 
of  the  air  as  it  passes  up  along  the  surface,  or  from  pressure 
against  the  biitts  of  the  shingles  or  slates,  is  too  slight  to  be 
of  any  consequence. 

Duchemin's  formula,  with  the  above  notation, 
P  .  2  sin'  i  ^  (1  +  sin^  i), 
gives  smaller  values  of  normal  wind  pressure. 

35.  Example  :  Steady  Load.— The  truss  of  Fig.  17  is 
supposed  to  be  under  the   action  of  wind  pressure  from  the 


24  EOOF-TEUSSES, 

left.  If  the  truss  is  67  feet  span,  and  the  height  jj  15  feet,  the 
angle  of  inclination  will  be  2-1°  7',  and  the  normal  wind  press- 
ure, interpolated  from  the  table,  will  be  21.8  pounds  per  square 
foot.  The  rafter  will  be  36.7  feet  long.  If  the  trusses  are  10 
feet  apart,  the  normal  wind  pressure  on  one  side  will  be 

36.7  X  10  X  21.8  =  8000  lbs. 

For  steady  load  of  slates,  boards,  rafters,  purlins,  and  truss, 
let  us  assume  11  pounds  per  square  foot  of  roof,  or 

36.7  X  10  X  2  X  11  =  8074  lbs.,  total  vertical  load. 

The  truss  is  here  drawn  to  a  scale  of  30  feet  to  an  inch,  and 
both  diagrams  are  drawn  to  a  scale  of  6000  pounds  to  an  inch. 
In  actual  practice  these  figures  should  be  much  larger,  the 
diagrams  showing  perhaps  1000  pounds  or  800  pounds  to  an 
inch. 

We  will,  in  the  present  case,  treat  the  two  kinds  of  external 
force  separately.  The  diagram  on  the  right  for  steady  load 
needs  no  description.  Each  supporting  force  will  be  4037 
pounds,  and  the  weights  at  the  joints  of  the  rafters  will  be, 
673  pounds  for  the  end  ones,  and  1346  pounds  for  each  of  the 
others.  The  above  weights  are  laid  off  on  a  vertical  load  line 
and  the  diagram  then  drawn.  The  stresses  in  the  various 
pieces  for  half  of  the  truss  are  given  in  the  table  to  follow, 
the  sign  -\-  denoting  compression,  and  the  sign  — ,  tension. 

36.  Wind  Diagram  ;  Reactions. — The  normal  pressure  of 
8000  pounds  distributed  uniformly  over  the  whole  of  the  left 
side  of  the  roof,  and  on  that  alone,  will  have  its  resultant,  shown 
by  the  dotted  arrow,  at  the  middle  of  that  rafter.  To  find 
the  supporting  force  on  the  right  we  may  take  moments  about 
the  left-hand  wall,  remembering  to  multiply  each  force  by  the 
lever  arm  drawn  perpendicular  to  its  direction  :  or 

AP  X  HT    =8000  X  HK, 
or 

AP  X  61.15  =  8000  X  18.35; 

■whence  A  P  =  2400  pounds,  and  A  H  =  5600  pounds. 


ROOF-TRUSSES.  25 

But  since  these  arms,  H  T  and  H  K,  are  proportional  to  the 
span  and  the  left  part  of  the  horizontal  tie  cut  off  by  the  re- 
sultant, an  easier  way  to  get  the  supporting  pressures  due  to 
an  inclined  force  is  to  prolong  this  force  until  it  cuts  the 
horizontal  line  joining  the  two  abutments,  when  the  two  reac- 
tions will  be  inversely  proportional  to  the  two  segments  into 
which  the  horizontal  line  is  thus  divided,  the  larger  force 
being  on  the  side  of  the  shorter  segment,  or,  for  ordinary 
pitches,  on  the  side  on  which  the  wind  blows. 

The  pressures  on  the  joints  will  be  2667  pounds  each  on 
IK  and  KL,  and  1333  pounds  each  on  HI  and  LM,  as  de- 
noted by  the  arrows.  Draw  m  h  by  scale,  equal  to  8000 
pounds,  so  inclined  as  to  be  in  the  direction  of  the  given 
forces,  that  is,  perpendicular  to  the  roof  ;  divide  the  reactions 
of  the  supports  by  means  of  the  point  a,  and  lay  off  the  joint 
forces  in  their  proper  order,  m  ?,  /  k,  k  i  and  /  h.  Before  going 
further  be  sure  that  the  external  forces  and  the  reactions 
follow  one  another  in  their  proper  order,  down  and  up  the 
load  line  ;  for,  through  heedlessness,  the  reactions  are  some- 
times interchanged. 

37.  Wind  Diagram;  Stresses. — Proceed  with  the  con 
struction  of  the  diagram  by  the  usual  rules,  remembering  that 
wind  alone  is  being  treated.  After  the  joint  K  L  has  given 
Ikcdel,  the  joint  EA  gives  eda/e.  Taking  next  the  apex 
L M,  and  passing  along  ml,le  and  e/,  we  find  that  there  will 
be  no  line  parallel  to  F  G,  since  g  m,  parallel  to  G  M,  will 
exactly  close  on  m,  the  point  of  beginning.  As  no  stress  passes 
through  F  G,  the  remainder  of  the  bracing  on  this  side  can 
experience  no  stress,  and  therefore  the  compression  g  m  affects 
the  whole  of  the  right-hand  rafter  while  the  tension  a/is 
found  in  the  remainder  of  the  horizontal  tie.  The  stress  tri- 
angle for  the  point  P  will  therefore  be  m  g  a  m.  That  the 
above  result  is  true  will  be  seen  if  we  notice  that  the  piece 
Q  K,  having  no  wind  pressure  at  its  upper  end,  can,  by  §  17, 
have  no  stress.  Then  it  follows  that  ES  is  now  free  from 
stress,  and  next  SG  and  lastly  GF,  all  by  §  17.     Further: 


26 


ROOF-TRUSSES. 


imagine  all  of  the  braces  in  the  right  half  to  be  removed  ;  it  is 
evident  that  the  right  rafter  is  a  sufficient  support  to  the  joint 
L  M,  conveying  to  the  wall  the  stress  g  m  which  compresses 
its  upper  end,  while  the  tie  A  F  keeps  the  truss  from  spread- 
ing. If  the  lower  tie  or  the  rafter  was  not  straight,  some  of 
the  braces  would  come  into  action,  as  will  be  seen  later. 

38.  Remarks. — At  another  time  the  wind  may  blow  on  the 
right  side.  Then  the  braces  on  the  right  will  be  strained  as 
those  on  the  left  now  are,  and  those  on  the  left  will  be  un- 
strained. The  wind  stresses  are  j)laced  in  the  third  column 
of  the  table.  As  in  this  truss  they  are  all  of  the  same  kind, 
in  the  respective  j^ieces,  as  those  from  the  steady  load,  they 
are  added  to  give  the  total  or  maximum  stresses.  The  force 
g  m,  being  smaller  than,  while  it  is  of  the  same  kind  as  I  e,  is  of 
no  consequence  ;  for,  with  wind  on  the  right,  M  G  would  have 
to  resist  a  stress  equal  to  I  e. 

A  combination  of  the  two  components  of  the  supporting 
forces  at  each  end,  as  shown  in  the  figure,  by  either  the 
parallelogram  or  triangle  of  force,  will  give  the  direction  and 
amount  of  each  reaction  from  the  combined  load.  Wind  on 
the  other  side  will  exactly  reverse  the  amounts  and  bring 
them  on  the  opposite  side  of  the  vertical  line. 


Table  of  Stresses  for  Fig.  17. 

Piece. 

steady  Load. 

Wind. 

Total. 

(  AB 

-  7520  lbs. 

10,440  lbs. 

17,960  lbs. 

Tie       ^AD 

—  6020 

7,160 

13,180 

/  AF 

-  4520 

3,900 

8,420 

EF 

-1830 

3,990 

5,820 

Braces  i  g^ 

—  1500 

3,280 

4,780 

+  1230 

2,670 

3,900 

[de 

+  1840 

4,000 

5.840 

(IB 

+  8240 

9.530 

17.770 

Kafter  ^  K  C 

+  7690 

9,530 

17,220 

(le 

+  5760 

6,550 

12,310 

If  the  truss  is  simply  placed  upon  the  wall-plates,  and 
either  of  the  supporting  forces  makes  a  greater  angle  with  the 


ROOF-TRUSSES.  27 

vertical  than  the  angle  of  repose  between  the  two  surfaces, 
the  truss  should  be  bolted  down  to  the  wall  ;  otherwise  there 
will  be  a  teudeucy  to  slide,  diminishing  the  tension  in  the  tie, 
perhaps  causing  compression  in  that  member,  and  changing 
the  action  of  other  parts  of  the  truss.  This  matter  will  be 
treated  of  further. 

If  the  weight  of  snow  is  also  to  be  provided  for,  it  may 
readily  be  done  by  taking  the  proper  fraction  of  the  stresses 
from  the  steady  load  and  adding  them  to  the  above  table. 

39.  Truss  with  Roller  Bearing ;  Dimensions  and  Load. 
— We  propose,  in  the  example  illustrated  by  Fig.  18,  to  con- 
sider the  truss  as  supported  on  a  rocker  or  rollers  at  the  end 
T,  where  the  small  circle  is  drawn,  to  allow  for  the  ex2Dansion 
and  contraction  of  an  iron  frame  from  changes  of  temperature. 
It  is  therefore  plain  that  the  reaction  at  T  must  alwavs  be 
practically  vertical.  The  truss  is  supposed  to  be  79  feet  8 
inches  in  span,  and  23  feet  in  height,  which  gives  an  angle  of 
30°  with  the  horizon,  and  makes  the  length  of  rafter  46  feet. 
It  would  be  proper  usually  to  support  the  rafter  at  more 
numerous  points;  but  our  diagram  would  not  then  be  so 
clear,  with  its  small  scale,  from  multiplicity  of  lines,  and  one 
can  readily  extend  the  method  to  a  truss  of  more  pieces. 

This  frame  supports  8  feet  of  roof,  and  the  steady  load  per 
square  foot  of  roof  is  taken,  including  everything,  as  14 
pounds.     The  total  vertical  load  will  then  be 

14  X  46  X  2  X  8  =  10,304  lbs., 

or  1717  lbs.  on  each  joint  except  the  extreme  ones. 

We  find,  from  the  table  of  §  34,  that  the  normal  pressure 
of  the  wind,  for  a  horizontal  force  of  40  j^ounds  on  the  square 
foot,  may  be  taken  as  26.4  pounds  per  square  foot  of  a  roof 
surface  inclined  at  an  angle  of  30°.  The  total  wind  pressure, 
normal  to  the  roof,  will  therefore  be 

26.4  X  46  X  8  =  9715  lbs., 
or   3238  lbs.    and  1619  lbs.   on  the  middle  and  end  joints 


38  ROOF-TEUSSES. 

respectively'  of  one  rafter.  The  truss  is  drawn  to  a  scale  of 
40  feet  to  an  inch,  and  the  diagrams  to  that  of  8000  j)ounds 
to  an  inch. 

40.  Diagram  for  Steady  Load. — The  diagram  for  steady 
load,  ha\dng  a  vertical  load  line,  is  the  one  above  the  truss, 
and  a  little  more  than  one-half  is  shown.  The  only  piece  at 
all  troublesome  is  G  F.  On  arriving  in  our  analysis  at  the 
apex  of  the  roof,  or  at  the  middle  joint  of  the  lower  member, 
we  find  three  pieces  whose  stresses  are  undetermined :  but  as 
we  have  reached  the  middle  of  the  truss,  we  know  that  the 
diagram  will  be  symmetrical,  and  therefore  that  gf  will  be 
bisected  b}'  a  ?.  In  the  case  of  an  unsymmetrical  load  we  can 
recommence  at  the  other  point  of  support  and  close  on  the 
apex.  The  stresses  caused  by  this  load  are  given  in  the  first 
column  of  figures  in  the  table  in  §  44,  compression  being 
marked  -j-,  and  tension  — .  If  it  is  thought  necessary  to  pro- 
vide for  snow,  in  addition  to  the  stresses  yet  to  be  found  for 
wind,  make  another  column  in  the  table,  of  amounts  properly 
proportioned  to  those  just  found. 

41.  Wind  on  the  Left;  Reactions. — Upon  turning  our 
attention  to  the  other  diagrams,  we  shall  find  that  the  rollers 
at  T  cause  something  more  than  a  reversal  of  diagram, — often 
a  considerable  variation  of  stress,  when  the  wind  is  on  differ- 
ent sides  of  the  roof.  Taking  the  wind  as  blowing  from  the 
left,  we  draw  the  diagram  marked  W.  L.  The  line  qm,  9715 
lbs.,  §  39,  is  di^dded  and  lettered  as  shown  for  the  four  loads 
at  the  joints  where  arrows  are  drawn.  The  resultant  of  the 
wind  pressure,  at  the  middle  point  of  the  rafter,  when  pro- 
longed by  the  dotted  arrow,  will  divide  the  horizontal  line  or 
span  in  the  proportion  in  which  the  load  line  should  be 
divided  to  give  the  two  parallel  reactions,  if  there  were  no 
rollers  at  T.  This  proportion,  for  a  pitch  of  30°,  is  2  to  1 ;  it 
locates  the  point  a',  and  gives  ma' =  64:77  lbs.,  and  a' q  = 
3238  lbs. 

But  the  reaction  at  T  must  be  vertical,  and  consequently 
only  the  vertical  component  of  a'  q  can  be  found  at  T,  while 


ROOF-TRUSSES.  29 

the  horizontal  component  of  a'  q  must  come,  through  the  lower 
member,  from  the  resistance  of  the  other  wall.  Therefore 
draw  a'  a  horizontally  and .  we  shall  get  a  q  as  the  vertical 
reaction  at  T,  while  ma,  to  close  this  triangle  of  external 
forces,  must  give  the  direction  and  amount  of  the  reaction 
atM. 

42.  Verification. — It  may,  at  first  sight,  strike  the  reader 
that  this  analysis  will  not  be  correct ;  for,  if  only  the  vertical 
component  is  resisted  at  T,  and  if  we  decompose  the  resultant 
of  the  wind  pressure  at  O,  where  it  strikes  the  roof,  into  two 
components,  we  get  results  as  follows : 

Vert.  comp.  of  9715  lbs.,  for  angle  30°  =  8414  lbs. 
Hor.       "       "  "  "         "       =4858  lbs. 

The  vertical  from  the  middle  point  of  the  rafter  will  divide 
the  span  at  \  M  T.  Therefore,  amount  of  vertical  component 
carried  at  T  =  2103  lbs.,  and  the  remainder  is  supported  at  M, 
with  all  of  the  horizontal  component.  But  take  next  into 
account  the  moment,  or  the  tendency  of  the  horizontal  com- 
ponent at  O  to  cause  the  truss  to  overturn.  It  naturally 
decreases  the  pressure  at  M  and  increases  that  at  T,  or,  in 
other  words,  the  couple  formed  by  the  horizontal  component 
at  O  and  the  equal  horizontal  reaction  at  M  with  an  arm  of 
half  the  height  of  the  truss  must  be  balanced  by  an  opposite 
couple,  com23osed  of  a  tension  at  M  and  an  equal  compression 
at  T,  with  a  leverage  of  the  span.  Making  the  computation 
of  this  tension,  or  compression  T,  we  have 

4858  X  11.5  =  T  X  79f,  or  T  =  702  lbs. 
2103  +  702   =  2805  =  i  of  8414  lbs. 

as  obtained  by  the  first  process. 

Still  another  way  to  find  the  supporting  forces  is  to  prolong 
the  resultant  until  it  intersects  the  vertical  through  T,  then  to 
draw  a  line  from  M  to  the  point  of  intersection,  and  finally  to 
draw  ma  and  qa  parallel  to  the  lines  from  M  and  T.  This 
method  depends  for  its  truth  on  the  fact  that  the  three  external 


30 


KOOF-TRUSSES. 


forces  wliicli  keep  tlie  truss  in  equilibrium,  not  being  parallel 
must  meet  in  one  point. 

43.  Diagram  for  Wind  on  Left. — Ha^-ing  completed  the 
triangle  of  external  forces,  and  laid  off  tlie  pressures  on  tlie 
joints,  we  can  readily  draw  tlie  diagram.  It  will  be  found,  as 
in  Fig.  17,  §  37,  that  braces  on  the  right  experience  no  stress, 
the  lines  gf  and  e  q  closing  the  polygon  which  relates  to  the 
joint  P  Q.  If  the  lower  tie  were  cambered  to  the  joint  D  C, 
we  should  find  a  stress  from  wind  in  E  F  and  C  D,  but  not  in 
B  C  or  C  E,  as  explained  in  §  37. 

Upon  combining  with  the  inclined  reaction  m  a  the  steady 
load  reaction  also  marked  m  a,  the  direction  of  the  resultant 
supporting  force  at  M  will  be  found  ;  and  it  may  be  so  much 
inclined  to  the  vertical  that  provision  against  sliding  on  the 
wall-plate  at  M  should  be  made.  The  stresses  given  by  this 
diagram  for  wind  on  the  left  are  found  in  the  table  to  follow, 
in  the  column  marked  W.  L,  It  will  be  seen  that  all  of  them 
agree  in  Jiincl  with  those  for  steady  load. 

44  Diagram  for  Wind  on  Right. — This  diagram  is 
marked  W.  Ft.     The  supporting  force  at.T,  while  still  vertical, 


Table  of  Stresses 

FOR  Fig.  18. 

Piece. 

steady  Load. 

W.  L. 

W.  R. 

fBS 
CR 

+  8570  lbs. 

5600  lbs. 

8480  lbs. 

+  6850 

5600 

6540 

Rafters  J  ?J9 

+  5700 

5600 

5880 

J-VCtrJ-L^i.  O              T     T3 

+  5700 

5880 

5600 

KO 

+  6850 

6540 

5600 

,LN 

+  8570 

8480 

5600 

LA 

-  7440 

11400 

0 

Tip         J  H  A 

^^^       Ida 

-  5450 

7050 

0 

-  5450 

4850 

2150 

BA 

—  7440 

4850 

6480 

EC 

+  1720 

0 

3800 

CE 

+  1520 

0 

3300 

EF 

—  1000 

0 

2150 

Braces  - 

FG 

-  2300 

2500 

2500 

GI 

-  1000 

2150 

0 

IK 

+  1520 

3300 

0 

KL 

+  1720 

3800 

0 

ROOF-TEUSSES.  31 

is  greater  in  amount  than  before.  If  diagram  W.  L.  has  been 
already  constructed,  the  reaction  at  T  can  be  taken  as  that 
portion  of  the  vertical  component  of  the  wind  pressure  not 
included  in  a  g  of  that  figure ;  that  is,  aq-\-ta  =  vertical 
component  of  qm  or  pt.  If  this  should  be  the  first  diagram 
drawn,  find  the  supporting  forces  in  one  of  the  three  ways 
given  above.  The  reaction  at  M  is  rightly  denoted  by  ap,  for, 
when  the  wind  is  on  the  right,  there  is  no  external  force  to 
di^dde  the  space  from  M  to  P. 

The  point  a  is  moved  considerably  from  its  place  in  diagram 
W.  L.,  and  this  change  affects  the  amounts  of  stress  in  the 
horizontal  member,  but  not  in  those  pieces  which  bear  similar 
relations  to  the  two  sides  of  the  truss  ;  in  other  words,  I P  and 
E  Q  interchange  stresses,  etc.  In  some  forms  of  truss,  how- 
ever, we  find  more  material  changes.  In  the  present  example 
it  happens  that  the  vertical  fg  strikes  the  point  a,  so  that  ip, 
the  stress  in  the  rafter,  coincides  with  ap,  the  reaction  at  M ; 
the  wind  on  the  right  consequently  causes  no  stress  in  L  A 
and  H  A.  The  stresses  from  this  diagram  are  found  in  the 
last  column  of  the  table. 

45.  Remarks. — There  is  no  need  to  tabulate  the  stress  in 
K  H,  if  that  in  I  G  is  given,  nor  gh,  ii  k  i  is  given.  Notice 
that  the  joint  K  G  or  C  F  gives  a  parallelogram  in  each 
diagram,  the  stress  in  K I  passing  to  G  H  without  change,  so 
that  the  diagonals  which  cross  may  be  considered  and  built 
as  independent  pieces.  It  will  be  seen  on  inspection  of  the 
table  that  the  combination  of  steady  load  with  wind  on  the 
left  gives  maximum  stresses  in  I P,  K  O,  L  N,  L  A,  HA,  T>  A, 
G I,  IK,  and  K L,  while  the  remainder,  with  the  exception  of 
F  G,  have  maximum  stresses  for  wind  on  the  right.  F  G  is 
strained  alike  in  both  cases. 

These  wind  diagrams  may  be  drawn  on  either  side  of  the 
line  of  wind  force,  as  in  the  case  of  steady  load,  by  changing 
the  order  in  which  the  supporting  forces  are  taken,  going 
round  the  truss  and  joints  in  the  opposite  direction. 
Although  there  exist  two  four-sided  spaces  C  and  K,  the 


32  KOOF-TRUSSES. 

structure  is  sufficiently  braced  against  distortion ;  for  these 
spaces  are  surrounded  by  triangles  on  all  sides  but  one. 

It  may  perhaps  not  be  amiss  to  suggest  again  how  to  deter- 
mine the  kind  of  stress  in  any  member  without  retracing  the 
whole  polygon  for  any  joint.  Notice,  from  the  load  line, 
whether  the  forces  were  taken  in  right-hand  or  left-hand  rota- 
tion. Read  the  letters  of  a  piece  in  that  order  with  reference 
to  the  joint  at  one  end  of  it ;  then  read  the  stress  in  the 
diagram  in  that  same  order,  and  it  will  show  the  direction  of 
the  stress  in  the  piece,  either  to  or  from  that  joint.  Thus 
diagram  W.  L.  is  written  in  left-hand  rotation ;  K  L  is  then 
the  reading  for  that  brace  at  its  loioer  end,  and  k  I  reads  down- 
ward or  is  thrust.  If  we  read  L  K,  it  must  apply  to  its  upper 
end,  and  I  k  acts  upwards  or  thrusts  against  the  joint  near  N. 

Wind  diagrams  for  the  truss  of  Fig.  21  can  now  be  drawn. 
The  apex  of  the  roof  can  be  treated  first,  and  the  stresses, 
obtained  in  the  dotted  lines,  can  then  be  transferred  to  the 
ends  of  the  upper  horizontal  member.  The  truss  proper  goes 
no  higher. 


CHAPTER  V. 

WIND  PRESSURE   ON  CURB   (OR  MANSARD)   AND  CURVED   ROOFS. 

46.  Truss  for  Curb  Roof;  Steady  Load  Diagram. — To 

have  a  definite  problem  we  will  assume  that  the  truss  of 
Fig.  19,  drawn  to  scale  of  20  feet  to  an  inch,  is  50  feet  in  span, 
that  the  height  to  ridge  is  20  feet,  to  hi23S  14|  feet,  and  that 
C  D  is  14  feet.  The  sides  K  B  and  G  E  are  practically  16f 
feet  long,  at  an  angle  of  60°  with  the  horizon,  so  that  their 
horizontal  projection  is  8^  feet.  The  upper  rafters  are  17^ 
feet  long,  and  therefore  make  an  angle  with  the  horizon  of 
18°  19'.  The  trusses  are  assumed  to  be  8  feet  apart,  and  are 
loaded  at  the  joints  only.  The  rafters  in  a  larger  truss  would 
commonly  be  supported  at  intermediate  points ;  but  more 
lines  would  make  our  diagrams  less  plain. 

The  steady  load  is  taken  at  12  pounds  per  square  foot  of 
roof  surface,  or 

(2  X  16t  +  3  X  17j)13  X  8  =  6560  lbs.,  total  load. 

The  joint  L  will  carry  one-half  the  load  on  KB,  or  800 
pounds  ;  the  joint  I K  will  carry  one-half  the  load  on  K  B  and 
one-half  of  that  on  I C,  or  800  +  840  =  1640  pounds  ;  IH  =  840 
-|-  840  =  1680  pounds,  etc.  These  weights  are  laid  off,  in  the 
diagram  marked  S.  L.,  from  I  to  /  by  a  scale  of  4000  pounds 
to  an  inch,  and  the  diagram  is  drawn.  It  shows  that  the 
rafters  are  in  compression,  marked  -\-,  and  all  the  braces  in 
tension,  marked  — . 

47.  Snow  Diagram. — In  treating  this  truss  for  snow  load, 
it  is  considered  that  K  B  and  E  G  are  too  steep  for  any  weight 
of  snow  to  accumulate  there,  as  whatever  fell  on  them  would 

33 


34  KOOF-TRUSSES. 

soon  slide  off.  Therefore  a  weiglit  of  12  pounds  per  horizon- 
tal square  foot,  for  the  upper  rafters  only,  is  taken  for  the 
maximum  snow  load,  and,  as  the  horizontal  projection  of 
I C  +  D  H  is  331  feet,  that  load  will  be 

12  X  33i  X  8  =  3200  lbs., 

laid  off  from  k  to  g,  in  the  diagram  marked  S.  The  end  por- 
tions, Jc  i  and  h  g,  are  each  800  pounds,  and  i  h  is  1600  pounds. 
The  division  into  two  equal  reactions  at  the  points  of  support 
gives  a.  This  diagram  much  resembles  the  other,  but  there 
is  one  point  worth  noticing;  the  lines  of  stress,  ic  and  Jid, 
cross  in  the  first  diagram,  but  do  not  in  the  second  ;  while  the 
reverse  is  the  case  with  ed  and  be.  The  result  is  that  the 
stress  of  C  D  is  reversed  by  the  maximum  snow  load,  and,  as 
this  stress  is  greater  in  amount  than  the  one  for  the  weight  of 
roof  and  truss,  C  D  will  be  a  compression  member  whenever 
such  a  load  of  snow  falls  on  the  roof ;  and  will  be  in  tension 
when  that  load  is  removed.  The  stresses  from  these  two 
diagrams  are  marked  on  the  truss  above  each  piece  on  its  left 
with  the  usual  signs.  This  strain  sheet  is  more  convenient 
than  the  table  of  §  44. 

48.  Wind  from  the  Left ;  No  Roller. — When  the  rafters 
do  not  slope  directly  from  the  ridge  to  the  eaves,  but  are 
broken  into  tAvo  or  more  planes  of  descent,  we  shall  have 
wind  pressures  of  different  directions  and  intensities  on  the 
two  portions,  I C  and  K  B.  From  the  table  of  wind  pressures, 
§  34,  we  see  that  the  intensity  of  pressure  on  K  B  will  be  40 
pounds,  and  on  I C  16.9  pounds,  normally,  per  square  foot  of 
roof.  The  total  pressure  on  KB  therefore  will  be  40  X  16f 
X  8  =  5333  pounds,  of  which  one-half  will  be  supported  at 
the  joint  L,  and  the  other  half  at  the  joint  J,  as  indicated  by 
the  two  arrows  perpendicular  to  K  B.  The  pressure  on  I C 
will  be  16.9  X  17^  X  8  =  2366  pounds,  or  1183  pounds  on 
each  joint. 

If  the  truss  has  no  rollers  under  it,  the  diagram  marked 
W.  L.,  I.  is  obtained.     On  a  scale  of  4000  pounds  to  an  inch, 


ROOF-TRUSSES.  35 

hi  =  ij  =  1183  pounds  ;  j k  =  kl  =  2667  pounds.  For  ij  and 
jk  may  be  substituted  ik,ii  desired,  the  resultant  of  these 
two  components  at  J. 

To  find  the  supporting  forces : — Prolong  the  resultants  of 
the  wind  pressure  from  the  middle  point  of  each  rafter  to 
intersect  the  span  L  F.  The  resultant  K  will  be  resisted  at 
L  and  F  by  two  reactions  parallel  to  it,  and  inversely  propor- 
tional to  the  two  segments  into  which  this  resultant  divides 
L  F,  as  shown  in  §  36.  The  same  will  be  true  for  the  result- 
ant I.  By  scale,  or  from  the  known  angles,  it  will  be  found 
that  resultant  K  cuts  L  F  at  16f  feet,  or  one-third  the  span, 
from  L,  and  that  resultant  I  cuts  it  at  22.4:  feet  from  the  same 
end.  Dividing  jl  at  ^  its  length,  we  have  la'  for  one  com- 
ponent of  the  reaction  at  L  and  a'j  for  one  component  of  the 

22  4 
reaction  at  F.     If  we  divide  lij  at  — -'-  of  its  length,  y  a"  will 

oU 

be  a  component  of  the  supporting  force  at  L,  and  a"li  at  F. 
By  drawing  the  parallelogram  a'j  a"  a  we  shall  bring  the  com- 
ponent reactions  for  each  wall  together,  and  shall  have,  for 
the  supporting  force  at  L,  or  LA,  la'  and  a' a,  or  their  result- 
ant I  a ;  and  for  that  at  F,  a  a"  and  a"h,  which  combined  give 
a  Jl,  properly  called  A  H  in  the  truss,  since  the  letters  from  F 
to  H  are  not  in  use  at  present.  Take  care  to  lay  off  the  com- 
ponent reactions  on  the  proper  ends  of  the  wind-pressure 
lines. 

The  polygon  of  external  forces,  when  there  is  no  roller 
under  the  truss,  is  therefore  h i,  i k,  kl,l a,  and  a h.  The  com- 
pletion of  the  diagram,  by  drawing  lines  parallel  to  the  several 
pieces,  will  be  easy  without  further  explanation.  That  the 
point  e  should  apparently  fall  on  i  k  is  accidental.  The  signs 
affixed  to  the  lines  will  enable  one  to  see  readily  that  the 
stresses  in  B  C  and  E  A  are  now  reversed,  the  pressure  I K 
obliging  us  to  use  a  strut  to  keep  that  joint  in  place.  The 
resultant,  however,  from  the  combined  stresses  in  E  A  is  still 
tension.  The  amounts  given  by  diagram  W.  L.,  I.  have  not 
been  placed  on  the  truss,  as  we  prefer  to  treat  it  from  another 


36  EOOF-TRUSSES. 

point  of  view.  Had  tliey  been  used,  it  would  be  unnecessary 
to  draw  a  diagram  for  wind  on  the  right,  for  the  different 
members  of  the  truss  would  exchange  stresses  symmetrically ; 
that  is,  AB  would  have  the  stress  of  EA,  and  E  A  that  of 
A  B ;  D  H  of  C  I,  etc.,  C  D  remaining  the  same. 

49.  Wind  from  the  Left;  Roller  at  Left. — If  rollers 
are  placed  at  L,  to  permit  of  movement  resulting  from  change 
of  temperature,  the  supporting  forces  will  be  modified,  LA 
becoming  vertical.  The  diagram  marked  W.  L.,  II.  shows  the 
effect  of  this  change.  So  far  as  drawing  the  lines  of  wind 
pressure  liijkl,  the  polygon  of  external  forces  will  be 
obtained  in  the  same  manner  as  before.  We  may  then  draw 
the  parallelogram  and  locate  the  point  here  marked  a' ;  then 
draw  a' a  horizontally,  and  we  shall  get  I  a,  the  vertical  reac- 
tion at  L,  equal  to  the  vertical  component  of  Za  of  the  figure 
just  j)receding. 

In  case  the  former  diagram  has  not  been  drawn,  a  readier 
way  to  determine  I  a  will  be  as  follows  : — Draw  li  I,  plainly  the 
resultant  of  hj  and  jl;  then,  having  prolonged  the  dotted 
arrows  at  I  and  K  until  they  meet,  draw  a  line,  parallel  to  hi, 
through  their  intersection.  This  line  will  give  the  position 
of  the  resultant  of  the  wind  pressures,  and  I  h  is  now  to  be 
divided  in  the  inverse  ratio  of  the  two  segments  into  which  the 
resultant  divides  the  span  LF.  The  point  of  division  will 
fall  at  a",  from  which  draw  horizontally  a"a,  and  the  reac- 
tion I  a  is  thus  determined.  This  method  will  not  answer  for 
finding  the  supporting  forces  if  they  are  both  inclined,  as  it 
will  make  L  A  and  A  H  parallel  to  one  another.  The  reac- 
tion at  L  being  I  a,  the  one  at  F  is  a  h,  requiring  the  resistance 
at  F  of  the  entire  horizontal  component  of  the  wind  pressure. 

A  comparison  of  the  two  W.  L.  diagrams  will  show  that  the 
stress  in  every  piece  is  changed  very  decidedly  in  amount, 
and  that  in  a  number  of  pieces  the  stresses  are  reversed  by 
rollers  at  L.  These  latter  stresses  are  marked  on  the  truss, 
at  the  right  of  each  piece. 


ROOF-TEUSSES.  37 

50.  Wind  from  the  Right. — When  the  wind  blows  from 
the  right,  the  diagram  marked  W.  K.  will  be  obtained.  The 
lines  ihgf,  representing  the  wind  j^ressures,  will  correspond 
in  value  with  hikl  of  the  preceding  figure,  and,  since  the  other 
diagram  has  been  constructed,  the  vertical  reaction  at  L  will 
now  be  obtained  by  drawing  the  horizontal  line  a'  a,  from  either 
the  angle  of  the  parallelogram  or  the  j^roper  point  of  division 
of  the  resultant  if,  so  as  to  give  a  i,  the  smaller  part  of  the 
vertical  component  of  the  wind  pressure  ;  that,  is  I  a  from  W. 
L.,  IL,  plus  a  {from  W.  K.,  equals  the  vertical  projection  of 
the  polygon  of  external  forces. 

51.  Results. — When  this  diagram  is  completed  by  the 
customary  rules,  a  comj^arison  of  it  with  the  one  preceding 
will  make  clear  the  effect  of  wind  on  different  sides.  The. 
stress  in  the  rafters  is  much  greater  when  the  wind  blows  on 
the  side  farther  from  the  rollers,  but  it  is  always  comj)ressive. 
The  forces  in  the  braces  are  all  reversed. 

The  weight  of  the  roof  and  truss  may  be  the  only  external 
force,  or  snow  may  be  added ;  and,  in  either  case,  the  wind 
may  also  blow  on  one  side  or  the  other.  Selecting  then  those 
stresses  which  may  exist  together,  we  find  the  maximum  tension 
and  compression  marked  below  each  piece.  The  rafters 
are  always  compressed,  and  A  B  is  always  in  tension.  The 
other  pieces  must  be  designed  to  resist  both  kinds  of  stress, 
although  the  compression  in  D  E  is  quite  insignificant. 

52.  Curved  Roof  Truss :  Example. — If  the  truss  has  a 
curved  exterior  outline,  the  pressure  of  the  wind  will  make  a 
different  angle  wdth  the  horizon  for  every  point.  But  there 
will  be  no  sensible  error  if  the  pressure  on  each  piece  is  as- 
sumed to  be  normal  to  the  curve  at  its  middle  point,  or,  what 
is  practically  the  same  thing,  perpendicular  to  the  straight 
line  joining  its  two  extremities.  Thus,  in  the  truss  of  Fig.  20, 
the  wind  pressure  on  C  T  is  taken  as  perpendicular  to  a 
straight  line  from  B  to  the  next  joint  in  the  rafter. 

The  span  of  this  truss,  drawn  on  a  scale  of  30  feet  to  an 
inch,  is  60  feet ;  height  at  middle  of  rafters  15  feet,  at  middle 


38  EOOF-TRUSSES. 

of  main  tie  6  feet.  The  curves  are  arcs  of  circles,  the  radii  of 
the  uj)per  and  lower  members  being  respectively  37^  feet  and 
78  feet.  The  rafters  are  spaced  off  at  intervals  of  11^  feet 
each  way  from  the  middle,  and  the  tie  is  di^ided  into  10^  feet 
lengths.  The  end  portions  will  diifer  slightly  from  these 
measures.  The  trusses  are  to  be  10  feet  apart.  From  the 
data,  radius  37^  feet,  and  half-chord  or  sine  5|  feet,  it  is  easy 
to  calculate  that  the  chord  of  the  first  piece  of  rafter  from  the 
middle  will  make  an  angle  with  the  horizon  of  8°  49^'.  The 
second  piece  will  be  inclined  three  times  as  much,  or  26°  28', 
and  the  last  five  times  as  much,  or  44°  6'.  The  intensity  of 
normal  wind  pressure  will  then  be,  when  interpolated  in  the 
table,  §  34,  8.6  pounds  per  square  foot  for  the  upper  length, 
23.7  pounds  for  the  next  length,  and  35.6  pounds  for  the  low- 
est piece.  Multiplying  these  intensities  by  11|-  X  10,  we  get 
989  ]30unds,  2725  pounds,  and  4094  pounds,  respectively,  repre- 
sented by  the  small  arrows,  as  if  concentrated  at  the  middle 
points  of  E,  D,  and  C.  The  steady  load  is  taken  at  a  small 
figure,  2300  pounds  per  piece  of  rafter,  to  allow  the  disturbing 
effect  of  the  wind  to  be  more  marked. 

The  diagonals  in  this  truss  are  light  iron  rods,  not  adapted 
to  resist  compression,  and  therefore,  if  a  compressive  stress 
would  occur  in  a  particular  diagonal,  in  case  it  were  alone  in  a 
panel,  we  substitute  the  other  diagonal,  which  will  then  be  in 
tension.  In  lettering  the  figure,  that  tie  which  is  required  for 
a  particular  distribution  of  load  is  supposed  to  be  present, 
and  the  other  diagonal  is  not  taken  account  of.  Thus,  in  the 
panel  through  which  the  dotted  arrow  is  drawn,  if  the  brace 
which  goes  from  the  top  of  O  P  to  the  bottom  of  Q  R  is  under 
stress,  it  will  be  called  P  Q,  while  the  rafter  will  be  Q  E  and 
the  bottom  tie  PA.  If  the  other  diagonal  is  strained,  the 
rafter  will  be  called  P  E  and  the  main  tie  Q  A. 

53.  Steady-Load  Diagram. — The  diagram  for  weight  of 
roof  and  truss  is  drawn  on  a  scale  of  8000  pounds  to  an  inch. 
The  vertical  load  line  is  i  b,  and  the  polygon  for  the  point  of 
support  ^  is  cb ate.     On  passing  to  the  next  joint  in  the  top 


ROOF-TRUSSES.  39 

or  bottom  member  we  find  three  pieces  "wliose  stresses  are 
unknown.  Both  diagonals  R  S  cannot  be  in  action  as  ties  at 
once ;  therefore  suppress  one,  for  instance  that  which  runs  to 
the  upper  end  of  S  T.  We  then  shall  have  only  two  unknown 
stresses  at  the  upper  joint,  and  can  draw  t  s'  and  s'  d.  The 
lower  joint  will  then  give  s't,  ta,  ar',  and  r's'.  But  r's'  will 
be  a  compressive  stress,  as  we  read  from  r'  to  s\  and  this 
diagonal  is  not  the  desired  one.  Taking  the  other,  and  trjdng 
the  lower  joint  first,  we  have  t ast,  and  the  uj^per  joint  then 
gives  dctsrd,  where  sr  is  tension.  Notice  that  change  of 
diagonal  aflfects  the  stresses  in  no  pieces  beyond  those  which 
bound  the  quadrilateral  or  panel  in  which  the  diagonal  is 
changed.  Analogy  will  rightly  lead  us  to  take  the  other  diag- 
onals which  slope  the  same  way,  that  is,  down  towards  the 
middle.  It  is  therefore  easy,  after  the  first  attempt,  to  decide 
which  diagonal  to  reject  and  which  to  retain. 

54.  Remarks. — If  d  r  had  been  slightly  more  inclined,  so  as 
to  strike  s,  no  diagonal  B  S  would  have  been  reqiiired  for  this 
distribution  of  load.  It  will  be  seen  that  the  stresses,  all  tensile, 
in  the  bracing  are  very  small  as  compared  with  those  in  the 
main  members,  a  fact  due  to  the  approximation  of  the  rafter 
outline  to  the  equilibrium  curve  or  polygon  for  a  load  dis- 
tributed as  in  this  case.  See  §  88.  If  the  outline  of  a  truss 
coincides  with  the  equilibrium  polygon  pertaining  to  a  certain 
distribution  of  load,  no  interior  bracing  will  theoretically  be 
needed  for  such  distribution ;  but  if  the  distribution  or  direc- 
tion of  the  external  forces  is  at  any  time  changed,  bracing 
will  be  called  into  action.  Further  discussion  of  this  subject 
comes  in  Parts  II.  and  III. 

The  length  of  hk,  etc.,  as  compared  with  H  K,  etc.,  shows 
the  necessity  of  drawing  the  truss  skeleton  on  a  large  scale, 
to  secure  parallelism  of  the  respective  lines  in  each  figure. 
As  a  slight  change  in  the  inclinations  of  the  rafter  and  lower 
tie  lines  will  change  the  magnitude  of  the  stresses  in  those 
pieces  quite  materially,  we  are  warned  by  the  appearance  of 
the  diagram  to  provide,  by  an  increase  in  size  of  these  pieces, 


40  ROOF-TKUSSES. 

against  such  a  cliange  in  the  truss  as  would  be  caused  by 
slight  errors  in  construction  or  by  deflection  under  the  load. 
Stress  diagrams  are  particularly  serviceable  in  this  way. 

55.  Wind  and  Steady  Load. — We  might  analyze  the  effect 
of  the  wind  separately  upon  the  truss,  but,  as  there  is  a 
likelihood  that  the  wind  will  reverse  the  stress  in  some  of  the 
diagonals  which  experience  tension  from  the  steady  load,  and 
that  we  shall  be  obliged,  therefore,  to  substitute  the  other 
diagonals  in  such  panels,  it  seems  better  to  draw  the  diagram 
for  the  wind  and  the  weight  of  the  roof  in  conjunction. 
Therefore  the  two  diagrams  marked  W.  R.  and  W.  L.  are 
drawn  for  the  maximum  force  of  wind  on  either  side,  com- 
bined with  the  weight  of  the  roof,  etc.  The  external  load  line 
hi  of  one  case  is  the  exact  reverse  of  ih  of  the  other.  An 
explanation  of  the  construction  of  W.  R.  will  suffice  for  both. 

When  the  wind  blows  from  the  right,  there  is  only  the 
steady  load  on  the  left  half  of  the  truss.  Beginning  therefore 
with  the  joint  at  I,  lay  off  vertically  hi  =  1150  pounds,  or 
one-half  the  load  on  H K ;  next  gh  =  2300  pounds,  load  at 
G  H,  and  so  on  to  F  E,  as  in  the  steady-load  diagram  already 
discussed.  At  FE  we  find,  in  addition  to  2300  pounds  verti- 
cal pressure,  an  inclined  force  perpendicular  to  the  tangent 
at  E,  or  to  the  chord  of  the  piece,  and  equal  to  one-half  of  989 
pounds,  the  wind  pressure  before  computed  for  E.  We  thus 
get  the  inclined  line  as  far  as  e  in  the  diagram.  The  joint 
DE  gives  de,  manifestly  made  up  of  the  other  half  of  989 
pounds,  of  the  vertical  2300  pounds  as  usual,  and  finally  of 
one-half  of  2725  pounds  from  the  next  length  of  rafter,  and 
perpendicular  to  it.  The  forces  for  the  remaining  joints  C  D 
and  B  C  will  be  plotted  in  the  same  manner,  and  we  therefore 
see  that,  commencing  at  B,  as  is  proper  for  this  load  line,  we 
lay  off  the  vertical  and  inclined  forces  in  regular  succession 
from  one  side  of  the  truss  to  the  other.  If  one  draws  a 
straight  line  from  c  to  d,  it  will  be  the  resultant  of  the  com- 
bined external  forces  at  C  D. 


EOOF-TRUSSES.  41 

56.  Reactions  and  Diagrams. — Connect  6  with  i  by  the 
dotted  line,  which  will  be  the  resultant  of  all  these  forces. 
As  the  resultant  of  the  dead  weight,  symmetrically  distributed, 
acts  in  the  line  of  the  vertical  O  P,  and  hence  through  the 
centre  of  curvature  of  the  rafters,  and  as  the  wind  pressures 
all  point  to  the  same  centre  of  the  circle,  the  resultant, 
parallel  to  bi,  must  pass  through  the  same  point.  Therefore 
draw  the  dotted  arrow  through  the  centre  from  which  the 
rafter  was  struck,  and  parallel  to  hi.  This  arrow  cuts  the  span 
B  I,  by  measurement,  at  25\  feet  from  B,  or  34f  feet  from  I, 
The  resultant  bi  scales  20,620  pounds.  If  the  sup23ortirjg 
force  at  B  were  parallel  to  this  resultant,  it  would  be  found 
by  taking  moments  about  I,  when  we  should  have 

Bx60  =  20,620  x34f;       or      B  =  11,943  lbs. 

Lay  off  this  force  from  b  to  a'.  If  rollers  are  placed  at  B, 
that  reaction  will  be  vertical,  and  the  horizontal  component 
of  a'  b  must  be  resisted  at  I.  Let  fall  b  a  vertically,  determin- 
ing the  point  a  by  drawing  a'  a  horizontally,  and  connect  i 
with  a.     The  two  supporting  forces  will  be  ia  and  ab. 

In  the  W.  L,  diagram  the  point  a'  comes  nearer  to  b  than  to 
i, — that  is,  the  quantity  just  obtained  now  applies  to  the 
point  of  support  I, — and  a  falls  very  near  to,  but  just  outside 
of  /,  in  the  prolongation  of  the  vertical  line. 

If  there  are  no  rollers  under  the  truss,  find  the  supporting 
forces  for  each  oblique  pressure  separately,  as  in  §  48.  The 
same  course  must  be  pursued  when  the  curve  of  the  rafters 
is  not  circular,  as  the  forces  will  not  then  meet  at  a  common 
centre.  Having  thus  completed,  in  either  case,  the  polygon 
of  external  forces,  the  remainder  of  the  construction  will  be 
made  as  in  any  example.  After  the  first  trial  to  ascertain  the 
proper  diagonal,  it  appears  that,  in  each  case,  the  diagonals 
all  slant  one  way  ;  so  that,  for  wind  on  one  side,  one  set  of 
diagonals  is  in  tension,  and  for  wind  on  the  other,  all  of  the 
other  set  are  strained. 


42  EOOF-TRUSSES. 

57.  Change  of  Diagonal. — The  effect  on  the  five  pieces  of 
a  panel,  top,  bottom,  two  sides  and  the  diagonal,  of  drawing 
the  diagram  so  as  to  give  compression  in  a  diagonal,  is  shown 
anew  in  the  W.  L.  figure  for  the  panel  P  Q.  Instead  of  op  and 
qr,  we  get  op'  and  q'r,  considerably  increased  in  amount  but 
the  same  in  kind ;  for  ep  and  aq  are  substituted  eq'  and  ap'y 
unchanged  in  kind,  but  having  practically  what  is  taken  from 
one  added  to  the  other ;  while  the  diagonal  stress  is,  as  we 
said,  reversed,  but  very  nearly  the  same  in  amount. 

It  might  be  practicable  to  deduce  some  rule  for  determining 
"beforehand  the  diagonal  which  would  have  the  desired  kind 
of  stress,  but  the  tentative  process  seems  easy.  We  find  it 
convenient  to  draw  the  lines  j^arallel  to  the  rafter  and  main 
tie  first,  as  ep  and  ap' ,  then  to  sketch  roughly  two  lines  for 
the  suspending  piece  and  diagonal,  see  whether  that  diagonal 
comes  in  tension,  and  finally  draw  the  right  ones  carefully. 

58.  Resultant  Stresses. — It  is  not  necessary  to  put  the 
signs  -f-  and  —  on  these  lines,  for  it  may  be  seen  that  all  the 
rafter  is  compressed,  the  whole  lower  member  extended,  and 
all  of  the  diagonals  are  in  tension,  as  well  as  all  the  suspend- 
ing pieces  except  O  P  and  Q  R,  which  are  compressed  a  trifle 
when  the  maximum  wind  comes  from  the  right.  Such  pieces 
are  easily  selected,  if  one  notices  that  op  and  g  r  in  the  W.  R. 
diagram  are  drawn  in  a  direction  oi:)posite  to  the  j)revailing 
one. 

The  stresses  are  given  in  the  following  table.  The  lengths 
of  rafter  are  denoted  by  a  single  letter.  The  pieces  of  the 
main  tie,  having  the  letter  A  in  common,  have  also  the  letters 
which  stand  before  the  stresses  in  the  proper  columns.  The 
inclination  of  the  diagonal  is  shown  by  the  sign  prefixed  to 
the  stress.  The  effect  of  the  wind  on  the  roller  side  is  to 
materially  reduce  the  stress  in  a  large  portion  of  the  main  tie. 
The  light  bracing  required  is  a  marked  feature  of  this  type  of 
truss,  and  the  predominance  of  tensile  members  favors  the  use 
of  iron  bars.  The  two  compressions,  marked  -j-,  are  too  in- 
significant to  require  an  increase  of  section. 


EOOF-TEUSSES. 

Table  of  Stresses  for  Fig.  20. 


43 


! 

S.  L. 

W.  R. 

W.  L. 

Max. 

+ 

fc 

12.600 

18.900 

16,200 

18,900 

D 

11,400 

17,500 

15,600 

17,500 

Rafters - 

E 

10,800 

15,000 

16,200 

16,200 

F 

10,800 

13,300 

17,900 

17,900 

G 

11,400 

12,700 

20,100 

20.100 

.H 

12,600 

13,100 

21,800 

21,800 

' 

K 

9,600 

K      5,500 

K 

19,500 

19,500 

L 

9,500 

L      5,500 

M 

18,000 

18,000 

Mam  Tie A  - 

N 

10,400 

N      7,200 

0 

16,000 

16,000 

Q 

10,400 

P      9,000 

Q 

14,200 

14,200 

s 

9,500 

R    10,900 

s 

12,300 

12,300 

__ 

T 

9,600 

T    12,800 

T 

12,300 

12,800 

fLM 

\900 

\ 1,800 

/ 1,800 

1    >-Q 

Diagonals...  -i  pQ 

"400 
/400 

"2,100 
"2,400 

"2,400 
"2,200 

^RS 

"900 

"2,200 

"2,100 

fKL 

1,200 

700 

1,200 

1,200 

MN 

1,000 

200 

900 

1,000 

Suspenders.  ^  OP 

900 

+  100 

700 

900 

1  QR 

1,000  ■ 

+    50 

1,000 

1.000 

LST 

1,200 

400 

1,600 

1,600 

If  the  designer  proposes  to  proportion  tlie  pieces  with  re- 
gard to  minimum  as  well  as  maximum  stresses,  he  can  readily 
select  the  former  from  the  table. 

If  a  fall  of  snow  is  supposed  to  be  uniformly  distributed 
over  the  roof,  the  increased  action  of  the  several  pieces  can 
be  easily  obtained  by  proportion  from  column  S.  L.  But,  if 
it  is  thought  that  the  inclination  of  the  portions  near  C  and  H 
is  too  great  to  permit  of  snow  accumulating  there,  a  diagram 
for  snow  should  be  drawn.  The  horizontal  projection  of  a 
piece  of  the  rafter  is  properly  taken  when  reckoning  a  snow 
load. 

We  think  the  reader  will  have  no  difficulty  in  drawing  dia- 
grams for  a  truss  of  similar  outline,  but  with  only  a  system 
of  simple  triangular  bracing. 


CHAPTEE  YI. 

TRUSSES  WITH  HOEIZONTAL  THRUST. 

59.  Scissor  Truss.— Wlien  it  is  desired  to  strengtlien  the 
rafters  in  a  roof  of  moderate  span  by  supporting  them  at  their 
middle  points,  a  simple  means,  often  employed,  is  to  spike  on 
a  piece  from  the  lower  end  of  one  rafter  to  the  middle  of  the 
other,  as  shown  in  Fig.  22.  The  two  pieces  may  or  may  not 
be  fastened  together  where  they  cross.  At  the  first  glance 
we  should  say  that,  to  draw  the  diagram,  we  must  lay  off  the 
load  line  ke,  di^dde  it  as  usual,  and  then,  beginning  at  the 
joint  E,  draw  a'h'  and  h'f,  parallel  to  AB  and  BE.  Next,  for 
the  joint  F  G,  we  should  get  the  lines  h'c'  and  c'g.  For  the 
apex  we  should  have  three  lines,  viz.,  h  g,  g  c\  and  a  line  from 
c'  parallel  to  C  H  to  strike  h.  There  is  evidently  something 
wrong  here.  If  we  start  from  the  other  point  of  support  K, 
we  obtain  the  remainder  of  the  diagram  in  dotted  lines,  and 
find  that  we  have  two  points  marked  c',  some  distance  apart, 
which  ought  to  come  together ;  we  also  have  two  conspiring 
forces,  gc'  and  he',  whose  vertical  components  ought  to  bal- 
ance hg. 

Abandoning  this  diagram  for  the  present,  let  us  start  at  the 
apex  of  the  roof,  where  we  may  feel  sure  that  there  are  but 
two  unknown  forces.  Taking  the  load  h  g  at  that  point,  draw 
the  full  lines  gfc  and  cli.  Next  for  the  joint  GF,  starting  with 
c g,  pass  down  (//and  draw/6  and  h c.  The  joint  H I  will  simi- 
larly give  the  figure  ihcdi.  Lastly,  the  joint  AC  will  add  ba 
and  ad  to  the  stresses  d c  ancl  c b.  To  close  the  polygon  for 
the  joint  E  we  must  now  supply  to  a  bfe  the  line  e  o,  which 
must  be  the  inclined  reaction  at  E,  required  to  keep  this  truss 

44 


ROOF-TRUSSES.  45 

frtjm  sliding  outwards  ou  the  wall-plates,  on  the  supposition 
that  the  points  of  meeting  of  two  or  more  pieces  are  true  joints 
(ones  about  which  the  parts  are  free  to  turn).  As  e  a  may  be 
decomposed  into  ea'  and  a' a,  the  force  a' a  is  called  the  hori- 
zontal thrust  of  the  truss,  which  may  be  resisted  by  the  wall 
or  by  a  tie-rod  from  E  to  K.  The  pieces  of  this  truss  are  all 
in  compression. 

60.  Horizontal  Thrust  or  an  Additional  Member 
Necessary. — That  the  truss  is  not  in  equilibrium  without 
this  inclined  or  horizontal  reaction  at  the  walls  is  seen,  if  we 
suppose  that  E  and  K  are  not  prevented  from  sliding  later- 
ally ;  the  joint  A  C  will  drop,  the  joints  F  G  and  H  I  will 
approach  one  another,  and  the  angle  at  the  apex  will  become 
sharper.  This  change  will  take  place  unless  the  above  or 
some  other  restraining  force  is  applied.  The  trouble  arises 
from  the  four-sided  space  C,  which  is  here  free  to  change  its 
form.  A  member  added  in  either  diagonal  of  this  space  will 
cure  the  evil.  One  from  the  apex  to  the  joint  C  A  will  plainly 
act  as  a  tie,  and  will  be  found  to  supply  the  missing  line  c'c' 
in  the  dotted  diagram  iirst  drawn.  From  this  diagram  we  see 
that  the  stresses  in  most  of  the  pieces  will  then  be  greater 
than  when  the  resistance  comes  from  the  wall.  A  strut 
between  the  joints  F  G  and  H I  will  also  make  the  truss 
secure ;  the  reader  can  try  such  a  diagram,  and  see  what 
pieces  have  their  stresses  reversed  by  the  change.  Either  of 
the  above  modifications  puts  the  truss  into  the  class  having 
vertical  reactions. 

61.  Remarks. — As  these  trusses  are  usually  made,  reliance 
against  change  of  form,  where  little  or  no  horizontal  thrust 
is  supplied  by  the  walls,  is  placed  upon  the  stiffness  of  the 
rafters,  which  are  of  one  piece  from  ridge  to  eaves,  and  on 
that  of  the  two  braces  ;  but  a  failure  to  get  a  good  horizontal 
resistance  from  the  walls  has  sometimes  resulted  in  an 
unsightly  sagging  or  springing  of  rafters  and  braces.  The 
bending  moments  on  these  pieces  are  due  to  the  horizontal 


46  EOOF-TRUSSES. 

tlirust.  Bending  moments  on  a  rafter  or  other  piece  will  be 
considered  later. 

It  is  worthy  of  notice  that  c  d  equals  h  a,  or  that  the  thrust 
is  constant  throughout  the  brace.  Two  members  crossing  as 
at  A  must  naturally  give  a  parallelogram  in  the  stress  dia- 
gram ;  the  component  of  the  load  at  H  I  which  starts  down 
the  brace  will  pass  to  E  without  being  affected  by  crossing 
the  other  brace ;  yet,  to  resist  the  tendency  to  sag  spoken  of 
above,  and  for  the  reason  that  the  braces  are  better  able  to 
resist  thrust  by  mutually  sta^-iug  one  another,  it  is  advisable 
to  spike  them  together  at  their  intersection. 

62.  Hammer-beam  Truss;  Curved  Members. — Another 
example  where  the  horizontal  tlirust  of  the  truss  against  the 
walls  must  be  ascertained  is  shown  in  Fig.  23.  This  frame  is 
called  a  hammer-beam  truss,  and  is  a  handsome  type  often 
employed,  in  this  country  and  abroad,  for  the  support  of 
church  roofs,  the  bracing  being  visible  from  below,  and  the 
spaces  containing  more  or  less  ornamental  work.  When  the 
church  has  a  clear-story,  the  windows  come  between  the  trusses 
at  B,  the  truss  is  supported  on  columns,  and  the  roof  of  the  side 
aisle  takes  up  the  horizontal  thrust.  If  there  are  no  side 
roofs,  the  main  walls  are  jDi'operly  strengthened  by  but- 
tresses. 

It  will  be  well  to  note  in  advance  that  a  curved  piece  in  a 
truss,  so  far  as  the  transmission  of  the  force  from  one  joint 
to  another  is  concerned,  acts  as  if  it  lay  in  the  straight  line 
between  the  two  joints.  The  curved  members  in  the  present 
example  are  the  quadrants  of  a  circle.  They  may  have  any 
other  desired  curve,  depending  somewhat  upon  the  pitch  of 
the  roof.  If,  now,  we  consider  the  point  of  support  B  P  of  the 
truss,  and  remember  that  the  curved  brace  A  O  transmits  the 
force  between  its  two  extremities  as  if  it  were  straight,  it  will 
be  evident  that  the  thrust  of  the  inclined  piece,  if  any  thrust 
exists  in  it,  must  have  a  horizontal  component  which  cannot 
be  neutralized  by  a  vertical  supporting  force  alone.  There- 
fore, in  addition  to  the  reaction  of  half  the  weight  of  the  roof 


EOOF-TRUSSES.  47 

and  truss,  there  must  be  supplied  by  the  wall,  assisted  per- 
haps by  a  buttress  or  a  side  roof,  a  certain  horizontal  thrust. 

63.  Amount  of  the  Horizontal  Thrust. — To  determine 
the  value  of  this  thrust : — Let  W  equal  the  weight  of  truss 
and  load.  We  have  nine  loaded  joints,  and  there  is,  there- 
fore, -JW  at  each  joint  except  the  two  extreme  ones.  The 
portion  213  maj  be  considered  a  small  truss,  like  Fig.  7, 
superimposed  on  the  lower  or  main  truss  4  6  2  3  7  5,  and  thus 
bringing  additional  loads  on  the  points  2  and  3.  If  then  we 
regard  the  main  truss  as  a  trapezoidal  truss,  and  consider 
that  the  pieces  L  A  and  Q  A  are  unnecessary  because  the  load 
is  the  same  on  the  two  halves  of  the  frame,  the  trapezoidal 
truss  will  be  4  2  3  5,  the  brace  4-2  being  made  up  of  an  assem- 
blage of  pieces.  L  A  and  Q  A  will  be  required  when  wind 
acts  upon  the  roof.  Considering  the  trapezoidal  truss  42  3  5 
alone,  the  joint  2  will  carry  a  load  equal  to  that  on  D  M,  E  K, 
and  F  I,  or  f  W,  the  joint  3  will  carry  the  same  amount,  while 
4  will  support  i  "VV  from  C  N,  and  5  the  remainder.  If  then 
we  lay  off  on  a  vertical  line  f  W,  for  the  load  on  2,  and  draw 
lines  parallel  to  2-4  and  2-3  from  its  extremities,  the  line 
parallel  to  2-3  will  be  the  stress  in  the  same,  and  will  also, 
since  the  load  is  vertical,  be  the  horizontal  thrust  of  the  foot 
of  the  compound  brace  2-4.  This  force  is  marked  H  in  the 
dotted  triangle  drawn  below  the  truss.  A  reference  to  §  25, 
Fig.  14,  may  aid  one  in  understanding  the  above. 

64.  Stress  Diagram. — We  now  have  the  data  for  the 
stress  diagram,  of  wliicli  one-half  is  shown.  For  the  point  4, 
or  B  P,  we  have  the  upward  supporting  force  bp  =  ^  W,  next 
pa  =  11,  the  horizontal  thrust  just  determined  of  the  wall, 
etc.,  against  the  joint,  a  o  parallel  to  the  line  of  action  of  AO, 
and  finally  o  h,  the  pressure  of  the  post  O  B  on  4.  The  result- 
ant oihp  and  p  a,  or  ha,  may  of  course  be  used  for  the  reac- 
tion of  the  wall.  Taking  next  the  joint  6,  we  have  ch  the 
load,  ho  the  thrust  of  B  O,  and  we  then  draw  o n  and  n  c.  The 
joint  C  D  gives  den  m d.  The  joint  M  A  already  has  the  lines 
m  n,  11 0  and  o  a ;  since  the  line  which  is  to  close  on  m  must  be 


48  EOOF-TRUSSES. 

parallel  to  L M,  and  a  is  already  vertically  over  m,al  can  have 
no  length,  and  there  is  no  stress  in  A  L,  as  before  assumed. 
Upon  taking  the  joint  D  E  we  find  also  that  no  stress  exists 
in  L  K.  The  reader  must  not  think  this  fact  at  variance  with 
the  value  H  which  was  said  to  exist  in  2-3  when  we  consid- 
ered the  trapezoid  alone  ;  the  triangular  truss  12  3  will  plainly 
cause  a  tension  in  2-3,  and,  with  this  distribution  of  load, 
such  tension  will  exactly  neutralize  the  compression  caused 
in  the  same  piece  by  4-2.  If  one  will  consider  the  truss  as 
loaded  at  6,  2, 1,  3,  and  7  only,  thus  doing  away  with  N  M,  K  I, 
IG,  etc.,  he  will  find  that  a  diagram  will  then  give  some  com- 
pression in  K  L. 

Another  method  of  treatment  will  be  applied  to  this  truss 
later,  §  75, 

65.  Different  Horizontal  Thrusts  Consistent  with 
Equilibrium. — In  studying  Fig.  22  we  saw  that  the  stresses 
in  G  C  and  C  H  were  determined  by  the  load  G  H,  and  that 
the  space  C  would  become  distorted  unless  a  horizontal 
thrust  of  a  definite  amount,  here  a'a,  was  supplied  by  the 
walls.  In  Fig.  23  also  the  same  things  are  true ;  the  trape- 
zoidal truss  4  2  3  5  requires  a  certain  horizontal  thrust  at  the 
points  4  and  5  to  balance  its  load ;  a  greater  or  less  thrust 
will  cause  the  truss  to  rise  or  fall,  so  long  as  L  A  and  Q  A  are 
neglected,  for  in  that  case  motion  can  freely  occur  at  joints  2 
and  3.  If,  however,  these  pieces  are  under  stress,  a  greater 
or  less  horizontal  thrust  may  be  applied,  the  truss  will  still 
be  in  equilibrium,  and  the  diagram  will  close.  Indeed  a  ver- 
tical reaction  is  a  supposable  one,  in  which  case  O  A  must  be 
without  stress.  The  same  statement  applies  to  Fig.  22,  if  one 
of  the  diagonals  of  the  space  C  is  put  in.  As  all  roof-trusses 
of  small  depth  in  their  middle  section,  as  compared  with  their 
total  rise,  have  a  tendency  to  spread  under  a  load,  and  hence 
to  thrust  against  their  supports,  their  diagrams  should  be 
drawn  for  a  moderate  amount  of  thrust  at  least,  if  it  is  desired 
to  have  them  maintain  their  shape  ;  and  the  supports  should 
be  able  to  offer  this  resistance,  or  a  tie  should  be  carried  across 


EOOF-TRUSSES.  49 

below.  Otherwise,  in  addition  to  the  sagging,  a  large  increase 
of  stress  is  likely  to  be  found  in  some  of  the  parts  as  a  result 
of  a  vertical  reaction.  The  determination  of  the  horizontal 
thrust  in  a  braced  frame  of  this  kind  is  not  very  simple,  but 
may  be  worked  out  by  a  method  given  in  Part  III,  "  Arches," 
Chap.  XII. 

66.  Proof. — That  such  trusses  are  in  equilibrium  under  a 
greater  or  less  amount  of  horizontal  thrust,  or  even  when  the 
reactions  are  vertical,  provided  the  pieces  are  able  to  with- 
stand the  resulting  stresses,  is  illustrated  by  Fig.  24.  Here 
the  load  C  B  is  taken  as  twice  D  C.  The  vertical  reactions  b  a 
and  ad  are  calculated  by  the  method  of  §  26.  The  diagram 
with  unaccented  letters  is  then  drawn  and  closed  as  usual. 
Next,  any  horizontal  thrust  a  a'  at  the  points  of  support  is 
assumed  and  the  diagram  with  accented  letters  is  drawn. 
This  diagram  also  closes.  The  reduction  of  all  of  the  stresses 
except  that  in  fg  is  most  marked.  We  see  from  these  cases 
that  only  when  the  truss  admits  of  deformation  by  the  distor- 
tion of  some  interior  space  such  as  C  of  Fig.  22,  or  R  of  Fig. 
27,  is  the  horizontal  thrust  determinate  by  the  method  of 
these  chapters ;  and  that  moderately  inclined  reactions  or 
the  tension  of  a  horizontal  tie  between  points  of  support  are 
favorable  to  a  reduction  of  the  stresses. 

Arched  ribs  of  a  nearly  constant  depth,  not  infrequently 
employed  in  railroad  stations  and  public  halls,  will  be  treated 
in  Part  III. 


CHAPTER  VII. 

FORCES  NOT  APPLIED  AT  JOINTS. 

67.  First  Diagram. — In  the  trusses  heretofore  treated  the 
leads  have  beeu  conceutrated  at  those  points  only  which  were 
directly  supported.  It  sometimes  happens  that  the  cross- 
beams or  purlins,  which  connect  the  trusses  and  convey  the 
weight  from  the  secondary  rafters  to  the  main  rafters,  rest 
upon  the  latter  at  points  between  the  joints.  Let  us,  in  Fig. 
25,  assume  that  a  load  rests  upon  the  middle  of  each  of  the 
upper  rafters.  If  we  neglect  the  bending  action  of  the  load 
E  G  u23on  the  rafter  and  proceed  as  usual,  we  consider  that 
one-half  of  the  load  E  G  will  be  supported  at  each  of  the 
joints  C  E  and  G  K,  and  similarly  for  the  load  K  M.  There- 
fore, having  laid  off  the  weights  and  the  two  equal  reactions 
of  the  walls  on  the  load-line  of  the  first  diagram,  we  may  in- 
crease the  loads  on  the  joints  C  E,  G  K,  and  M  O  by  the  new 
points  of  division,  and  complete  this  diagram,  taking  first  B, 
then  the  next  joint  on  the  inside,  and  then  the  outside  one. 
It  will  be  noticed  that  all  of  the  pieces  except  the  rafters  are 
ties. 

68.  Supplying  Imaginary  Forces. — This  diagram  gives 
but  one  stress  along  the  whole  of  the  upper  rafter  ;  but  it  is 
plain  that  the  vertical  force  E  G  must  have  a  component  along 
the  rafter  and  cause  a  different  stress  to  exist  in  E  T  from 
what  exists  in  G  T.  If,  however,  we  suppose  a  joint  to  be  at 
E  G,  the  transverse  component  of  E  G  will  cause  it  to  yield, 
as  there  is  no  brace  beneath  to  hold  it  in  place.  To  secure 
equilibrium  here  we  may  sujjply  an  imaginary  force  EF, 
shown  by  the  dotted  line,  equal  and  directly  opposed  to  this 

50 


ROOF-TRUSSES.  61 

transverse  component.  Tliis  imaginary  force  will  take  the 
place  of  a  perpendicular  strut,  will  steady  the  joint,  and  will 
leave  the  longitudinal  component  to  affect  the  rafter.  But 
the  transverse  component  of  FG  actually  gives  a  pressure  at 
the  joints  C  E  and  G  K,  while  the  imaginary  force  E  E,  just 
added,  will  lift  the  ends  of  this  rafter  by  the  same  amount ; 
therefore  we  must  restore  the  pressure,  and  the  equilibrium 
of  the  rafter  F  T  as  a  whole,  by  adding  imaginary  forces,  each 
one-half  of  E  E,  at  C  D  and  G  H.  This  added  sjstem  of  forces 
cannot  interfere  with  the  stresses  in  any  other  pieces,  for  they 
balance  by  themselves.  Treat  the  similar  load  KM  in  the 
same  way. 

69.  Second  Diagram. — In  the  second  diagram  the  two 
supporting  forces,  pa  and  ah,  are  each  equal  to  one-half  the 
total  load.  Lay  off"  6  c  as  before  ;  draw  the  dotted  line  c  d,  equal 
and  parallel  to  the  first  imaginary  force  C  D  ;  then  de  vertical, 
as  before  ;  then  ef,  equal  to,  and  in  the  direction  of  E  F  ;  then 
fg,  and  so  on,  arriving  finally  atp,  as  usual. 

The  construction  of  the  rest  of  the  diagram  presents  no 
difficult}^ ;  the  joints  are  taken  in  the  same  order  as  before, 
and,  when  we  have  more  than  one  external  force  on  a  joint, 
we  take  them  in  succession,  in  the  order  first  observed  for  the 
external  forces.  When  we  reach  the  upper  rafters,  we  find 
that  g  falls  on  the  line  et;  etis,  greater  and  gt  in  less  than  the 
line  for  the  same  piece  in  the  first  diagram. 

70.  Comparison  of  Results. — Thus  it  appears  that  the 
first  diagram  gives  the  stress  which  would  exist  in  the  whole 
length  of  the  rafter  E  T  G,  if  the  load  E  G  were  actually  at  its 
extremities ;  but,  being  at  its  middle  point,  one-half  of  the 
longitudinal  component  of  EG  goes  to  diminish  the  compres- 
sion otherwise  existing  in  G  T,  and  the  other  half  to  increase 
the  compression  in  E  T.  A  comparison  of  the  two  diagrams 
will  also  show  the  truth  of  the  former  statement,  that  the 
system  of  imaginary  forces  does  not  affect  any  of  the  truss 
outside  of  the  particular  pieces  to  which  it  may  be  applied. 
It  is  still  necessary  to  provide  for  the  bending  action  of  the 


52  ROOF-TRUSSES. 

transverse  portion  of  F  G,  or  a  force  equal  and  opposite  to  E  F 
upon  the  rafter,  considered  as  a  beam  extending  from  hip  to 
apex,  a  joint  of  course  not  being  made  at  E  G.  This  subject 
will  be  treated  in  Chapter  IX. 

71.  Remarks. — If  the  action  of  the  wind  upon  this  truss  is 
considered,  it  will  be  seen  at  once  that  no  special  treatment  is 
needed ;  for  the  wind  pressure  is  normal,  and  the  addition  of 
the  opposite  force  EE  at  once  balances  the  force  on  this 
joint,  and  transfers  it  to  the  ends  D  and  H  as  the  first 
analysis  did.  The  bending  action  on  the  rafter  must,  how- 
ever, be  provided  for. 

The  treatment  of  loads  or  forces  not  directly  resisted,  at 
above,  is  given  by  Mr.  Bow  in  his  "  Economics  of  Construc- 
tion," and  may  be  applied  to  frames  where  one  or  more  of  the 
internal  spaces  are  not  triangles,  but  quadrilaterals.  If  such 
spaces  are  not  surrounded  by  triangular  spaces  on  at  least  all 
sides  but  one,  the  truss  is  liable  to  distortion,  unless  the  re- 
sistance of  some  of  the  pieces  to  bending  or  the  stiffness  of 
the  tJworetical  joints  is  called  into  play.  A  use  of  this  treat- 
ment at  many  points  in  the  same  diagram  will,  however,  be 
apt  to  make  confusion. 

Another  application  of  imaginary  forces,  where  a  bending 
moment  exists,  will  be  made  at  the  close  of  the  next  chapter. 


CHAPTER  VIII. 


FECIAL     SOLUTIONS. 


72.  Reversal  of  Diagonal. — Difficulty  is  sometimes  ex- 
perienced iu  completing  the  diagram  for  a  truss  because, 
after  passing  a  certain  point,  no  joint  can  be  found  where  but 
two  stresses  are  unknown ;  while  yet,  judging  from  the 
arrangement  of  the  pieces,  the  stresses  ought  apparently,  to 
be  determinate.  Such  a  case  was  found  in  Fig.  11,  and  was 
solved  in  §  20  by  what  might  be  called  the  law  of  symmetry. 
A  method  of  more  general  application  to  these  cases  is  what 
may  be  styled  Reversal  of  a  Diagonal. 

It  has  been  pointed  out  alreadj^  that,  if  any  quadrangular 
figure  in  a  truss  is  crossed  by  one  diagonal,  the  other  diagonal 
of  the  quadrangle  may  be  substituted  for  the  former  without 
affecting  the  stresses  in  any  pieces  except  those  which  make 
up  the  quadrangle.  See  §§  26  and  53.  It  will  be  found  that 
such  a  change  often  reduces  the  stress  in  one  or  more  pieces 
of  the  quadrangle  to  zero,  and  thus  makes  the  truss  solvable 
graphically.  It  will  be  well,  if  the  reader  fails  to  distinguish 
readily  the  altered  truss  from  the  original  one,  to  temporarily 
erase  from  a  pencil  sketch  the  pieces  thus  rendered  super- 
fluous, or  to  draw  the  truss  anew  with  the  proper  changes  as 
has  been  done  in  Figs.  26  and  27.  The  modified  truss  will 
then  be  easily  analyzed,  and,  when  the  old  members  are 
restored,  enough  stresses  will  be  known  to  make  the  final 
solution  practicable. 

73.  Example.— This  method  will  first  be  applied  to  the 
roof-truss,  Fig.  26,  of  a  railroad  station  at  Worcester,  Mass. 
The  span  of  this  roof  is  125  feet ;  entire  height,  wall  to  apex, 

53 


54  ROOF-TRUSSES. 

45  feet ;  camber  of  main  tie  8  feet ;  rafter  divided  into  six 
equal  panels  ;  trusses  50  feet  apart. 

Under  steady  load  the  tie  bars  S  T,  T  U,  U  W,  WS,  whicli 
cross  the  centre  line  of  the  truss,  will  be  without  stress,  as  in 
Fig.  14,  §  25.  Indeed,  as  these  two  centre  ties  are  indepen- 
dent of  one  another,  but  one  can  be  in  action  at  a  time,  as, 
for  instance,  S  W  and  T  U  when  the  wind  is  on  the  left  side. 
If  we  begin  our  diagram  from  B  with  chak c,  we  meet  with 
no  difficulty  until  we  have  passed  the  joint  E  F,  for  which  we 
drew  fen  op/.  At  either  of  the  next  joints  are  three  un- 
known stresses.  As  all  stresses  are  determined  up  to  the 
piece  P  Q,  change  the  diagonal  Q  II  in  the  adjoining  quadri- 
lateral from  the  position  of  the  full  line  to  the  dotted  one. 
Then  the  joint  F  G,  as  seen  in  the  sketch  below,  will  give  us. 
gfpg'g.  As  the  full-lined  diagonal  has  been  removed,  the 
joint  R  W  has  disappeared  ;  for,  if  three  supposed  forces  are 
in  equilibrium  at  one  point,  §  17,  and  two  of  them  act  in  one 
line,  the  third  force  must  be  zero,  and  II S  therefore  can  have 
no  stress.  The  stress  in  SW  will  also  be  zero  unless  it. 
resists  wind  on  the  left,  and  the  stress  in  S  T  is  then  zero. 
In  either  case  we  can  draw  h  g  g'r'h  for  the  upper  joint,  and  then 
find  a  w  and  w  r',  if  it  exists,  at  the  lower  joint.  The  dotted 
peak  is  not  in  the  main  truss,  but  in  the  jack-rafters  which 
transfer  their  load  to  G  H  and  H I ;  if  one  prefers,  he  may 
put  a  load  at  the  peak  and  draw  the  triangle  of  forces  for 
that  point. 

After  using  the  above  expedient  on  the  other  half  of  the 
truss  also,  if  the  load  is  unsymmetrical,  we  replace  the 
reversed  diagonal  and  find  the  true  stresses  in  the  pieces 
affected  by  the  change,'  ^the  diagonal  and  the  four  sides  of  the 
containing  quadrilateral.  Hence  we  may  draw  ^oa?^^'^  for 
the  lower  joint  or  hgrsli  for  the  upper  joint,  and  finally 
gfV  g'^g  ^or  '^^  left-hand  joint  of  the  quadrilateral. 

74.  Polonceau  Truss.— The  left  half  of  Fig.  29  is  the  same 
as  Fig.  11.  It  will  be  remembered  that  we  were  stopped  at 
the  piece  D  E  of  Fig.  29  by  having  three  unknown  stresses  at 


ROOF-TKUSSES.  55 

either  end.  Change  the  full  line  E  F  to  the  dotted  one.  The 
stress  in  F  G  at  once  becomes  zero,  as  did  K  S  in  Fig.  26. 
We  may  now  find  the  stresses  in  D  E  and  E  L  at  the  joint 
K  L  ;  in  dotted  E  F  and  G  M  at  joint  L  M,  and  in  A  H  and 
H  F  at  the  lower  joint.  Then  the  diagonal  may  be  replaced 
and  the  stresses  in  D  E,  E  F,  F  G,  E  H,  and  F  L  rectified. 
The  right  half  of  Fig.  29  may  be  similarly  solved  by  revers- 
ing the  diagonal  P  Q,  which  change  makes  the  stress  in 
O  P  zero. 

75.  Hammer-Beam  Truss,  by  Reversal  of  Diagonals. 
— The  hammer-beam  truss  of  Fig.  27  difters  from  that  of  Fig. 
23  by  the  omission  of  the  vertical  in  the  space  R.  As  pointed 
out  in  §  66,  this  omission  renders  the  horizontal  thrust  of  this 
truss  definite.  In  attem23tiug  to  draw  a  diagram,  however, 
we  cannot  apparently  begin  at  the  wall  until  we  know  the 
horizontal  thrust,  and,  if  we  begin  at  F  G,  we  soon  meet  with 
joints  where  three  unknown  forces  are  found.  The  method 
of  the  preceding  sections  will  first  be  applied  to  the  right 
half.  Draw  gfr  for  the  upper  joint,  ligrsTi  for  joint  GH, 
and/eg' 7'/ for  EF.  As  joints  HI  and  RA  are  now  insoluble, 
draw  dotted  T  W  for  the  full-lined  diagonal  T  W,  and  do  the 
same  with  X  Y.  The  truss  will  thus  be  changed  to  the  form 
of  the  sketch  below.  For,  since  T  A  and  Y  A  act  in  the  same 
straight  line  (shown  dotted  on  left  half  of  truss),  the  stress  in 
W  X  is  now  zero,  and  T  A  and  Y  A  have  the  same  stress. 
Further,  at  joint  K  L  there  remain  K  Y,  Y  L,  and  the  exterior 
force  or  load  K  L,  which  latter  acts  in  the  vertical  line  Y  L  ; 
hence  the  stress  in  K  Y  is  now  zero,  and  Y  L  carries  K  L 
only.  We  can  therefore  draw  ihst'i  for  joint  HI,  kit'iv'h 
for  joint  I K,  lu't's  rq  .  .  .  aio'  for  joint  A  R,  and  I  k  iv'a  I  for 
the  abutment.  The  reaction  a  I,  being  thus  determined,  can 
be  used  to  draw  the  diagram,  as  in  Fig.  23.  The  diagram  for 
the  left  half  of  the  truss  is  given  in  full  lines,  and  it  may  be 
seen  that  A  P  and  A  T  are  now  useful. 

76.  Method  of  Trial  and  Error.— Where   the  unknown 
stress  in  but    one   piece  apjDears  to  stand    in  the  way  of  a 


56  KOOF-TKTJSSES. 

solution,  the  diagram  may  sometimes  be  drawn  witli  com- 
parative ease  by  trial.  Tlius,  in  the  left  half  of  Fig.  27, 
we  may  assume  the  value  of  the  horizontal  thrust  or  of  the 
stress  in  P  Q  and  proceed  with  the  diagram.  Upon  its  failing 
to  close,  we  can  change  the  assumed  quantity  and  try  again. 
Thus,  beginning  at  the  apes,  draw  gfrg,  feqrf,  and 
hgrsh;  then  assume  qp'  and  its  equal  st'.  The  middle 
joint  will  give  t'srqp'a't';  the  joint  DE,  p'qedo'p',  etc.;  and 
finally  the  horizontal  line  from  n'  will  fail  to  meet  a  line 
parallel  to  A  M  on  the  load  line,  to  give  m  h  in  the  post.  It 
is  evident,  upon  a  slight  inspection,  that  qp'  is  too  long. 
The  reader  will  find  that  he  can  soon  bring  the  diagram  to  a 
closure  by  diminishing  qp' . 

By  the  use  of  such  apjjroximations  one  of  necessity  loses 
that  check  on  the  accuracy  of  the  diagram,  of  having  it  close 
with  reasonable  exactness. 

Fig.  30,  in  case  one  or  the  other  of  the  dotted  diagonals 
is  used,  will  serve  as  an  example  for  the  practice  of  the  pre- 
ceding suggestions.  Which  diagonal  tie,  if  either,  will  be 
needed  for  wind,  and  which  for  steady  load  ? 

77.  Example. — We  will  close  this  branch  of  the  subject 
with  an  example  which  will  introduce  one  or  two  new  points 
in  addition  to  a  combination  of  principles  heretofore  illus- 
trated separately.  The  example  shows  the  capabilities  of 
this  method  in  handling  complex  problems.  The  structure 
drawn  in  Fig.  28  is  to  be  treated  as  a  whole  in  its  resistance 
to  wind  pressure. 

The  steady-load  diagram  would  present  no  difficulty.  The 
truss  is  carried  upon  columns  which  are  hinged  at  their 
lower  ends  B  and  P,  each  being  connected  by  a  pin  to  its 
pedestal.  The  brace  at  E  is  therefore  necessary  to  prevent 
overturning.  The  proportions  of  the  frame  are  as  follows : 
Distance  between  columns,  76  ft.;  AC  =  15  ft.;  Q  R  =  7  ft.; 
camber  of  lower  tie,  3  ft.;  1-A  =  19  ft.;  height  of  space 
1  =  16  ft.;  of  Y  =  7  ft.;  extreme  height,  ground  to  peak,  48  ft. 


KOOF-TEUSSES.  67 

Distance  between  trusses,  12  ft.   Scale  40  ft.  =  1  in.    Scale  of 
diagram,  8000  lbs.  =  1  inch.     No  wind  on  C. 

Wind  pressure  on  main  roof,  12,000  lbs.  =  hj;  therefore /gr, 
gh,  etc.,  =  3000  lbs.;  wind  pressure  on  KX=3360  Ibs.^J-lO; 
on  L  Y  =  3500  lbs.  =  10-m.  The  dotted  arrows  are  resultants 
of  wind  pressure  on  the  sloping  surfaces.  By  moments 
about  P,  or  by  proportion  of  segments  of  span  BP,  as 
in  §  48,  we  find 

that  8368  lbs.  of  bj  is  carried  at  B,  and  3633  lbs.  at  P. 
that    940    "     "    10-m      "        "  "      "    2460    "     "  " 


9308  lbs.  =  6-9  "        "   "      "    6093    "     "  " 

The  horizontal  force,  y-10,  at  K,  may  be  supposed  to  be 
resisted  equally  at  each  point  of  support,  since  the  two  posts 
will  be  alike.  Hence  jk  =  9-a'  =  UJ-10)  =  1680  lbs.  is 
carried  at  B.  The  moment  of  this  horizontal  force  K  about 
B  or  P,  tending  to  overturn  the  frame,  or  the  couple  formed 
by  K  and  the  equal  reaction  in  the  line  P  B,  will  cause  an 
increased  upward  vertical  force  at  P  and  an  equal  downward 
force  or  diminished  pressure  at  B.     Its  value,  §  42,  will  be 

^  =  1760  pounds  =  a' a.     The   reaction  at   B  must 

76  ^ 

balance  the  components,  b-9,  9-«',  and  a'-a,  and  hence  will 
be  a  b.  The  reaction  at  P  will  then  be  m  (or  p)  a,  which  may 
be  checked  in  detail,  if  desired. 

The  reaction  ab,  at  B,  will  now  be  decomposed  into  its 
vertical  and  horizontal  comjjonents  ac  and  cb.  The  piece  AC 
can  resist  a  c  as  a  strut  or  post,  but  must  carry  c  b  5900  lbs.  by 
acting  like  a  beam.  Were  there  a  real  joint  at  D  the  struc- 
ture would  fall.  It  is  therefore  necessary  to  make  the  post 
of  one  piece,  or  as  one  member  from  B  to  R.  The  magni- 
tude of  the  horizontal  force  at  F  caused  by  the  5900  lbs.  of 
horizontal  force  at  B  will  be  in  the  ratio  of  the  two  segments 
of  the  column  (beam)  or  as  15  to  7,  or  12,643  lbs.  These  two 
forces  must  be  balanced  at  D  by  a  force  equal  to  their  sum. 


58  ROOF-TRUSSES, 

or  18,54i3  lbs.  As  in  §  68,  Fig.  25,  tliis  beam  action  of  tlie  post 
must  be  neutralized,  before  the  diagram  can  be  drawn,  as 
these  diagrams  take  no  account  of  bending  moments,  for 
which  see  Chap.  IX. 

We  therefore  apply  at  B  0  the  imaginary  horizontal  force 
be  =  5900  lbs.,  opposed  to  the  direction  of  the  reaction,  and 
leaving  only  a  c,  the  vertical  component,  which  is  balanced  by 
the  post;  at  CD  we  apply  ccZ  =  18,543  lbs.;  and  at  EF,  we 
add  ef  =  12,643  lbs.  The  sum  of  these  three  imaginary  hori- 
zontal forces  being  zero,  the  stresses  in  the  truss  are  not  dis- 
turbed. The  same  steps  must  be  taken  at  P,  the  horizontal 
forces  mn,  no,  and  op  being  obtained  by  the  same  process 
from  the  horizontal  component  po  ol  the  reaction  p  a. 

The  load  line  therefore  finally  becomes  bed efg hikl m n op, 
the  force  D  E  being  shifted  laterally  as  shown,  and  i  k  being 
the  resultant  of  ij  andjZ:;.  The  stress  in  D  Q  is  readily  ob- 
tained by  drawing  deq.  Then  the  point  D  of  the  post  gives 
the  figure  acdq  r  a,  determining  the  stresses  in  the  upper 
part  of  the  post  and  the  brace  R  A.  The  remainder  of  the 
diagram  presents  no  difliculty. 

The  column  must  be  designed  to  resist  the  large  bending 
moment  to  which  it  is  liable,  as  well  as  the  thrust  q  r.  For 
bending  moments,  etc.,  see  the  next  chapter,  and  also  Part  11. 
As  this  structure  is  supposed  to  be  open  below,  the  lower 
member  should  be  adapted  to  resist  such  compression  as  may 
come  upon  it  from  the  tendency  of  a  gust  of  wind,  entering 
beneath,  to  raise  the  roof. 


CHAPTEK  IX. 

BENDING  MOMENT  AND   MOMENT   OF    RESISTANCE, 

78.  Load  between  Joints. — Having  treated  of  the  action 
of  external  forces  upon  a  great  variety  of  trusses,  we  propose 
now  to  investigate  the  graphical  determination  of  the  bending 
moments  which  arise  from  the  load  on  certain  pieces,  and  of 
the  stresses  due  to  the  moments  of  resistance  by  which  the 
bending  moments  must  be  met. 

To  recapitulate  some  statements  of  earlier  chapters  : — In 
case  the  transverse  components  of  the  load  upon  a  portion  of 
a  rafter,  or  other  piece  of  a  truss,  are  not  immediately  resisted 
by  the  supporting  power  of  some  adjacent  parts,  or,  in  other 
words,  unless  the  load  on  a  structure  is  actually  concentrated 
at  the  several  joints,  such  transverse  components  will  exert  a 
bending  action  on  the  portion  in  question,  and  the  additional 
stress  thus  caused  in  the  piece  may  be  too  great  to  be  safely 
neglected.  Further,  in  case  the  piece  makes  any  other  than  a 
right  angle  with  the  line  of  action  of  the  load,  or  has  an 
oblique  force  acting  upor  it,  the  stress  along  it,  given  by  the 
diagram,  will  be  less  than  the  maximum,  and  will  generally  be 
the  mean  stress.  Lastly,  in  case  a  piece  is  curved,  a  bending 
moment  will  be  exerted  upon  it  by  the  force  acting  along  the 
straight  line  joining  its  two  ends,  this  bending  moment  being 
a  maximum  at  the  point  where  the  axis  or  centre  line  of  the 
piece  is  farthest  removed  from  the  line  drawn  between  its  ends. 

79.  Example. — To  illustrate  the  former  statements  by  a 
simple  example  : — Suppose  the  rafters  A  C  and  B  C,  Fig.  31, 
to  be  loaded  uniformly  over  their  Avhole  extent.  Let  us 
assume,  in  the  first  place,  that  the  tie  AB  is  not  used,  but 

59 


60  KOOF-TRUSSES. 

that  the  thrust  of  the  rafters  is  resisted  by  the  walls  which 
carry  the  roof.  Consider  the  piece  A  C.  Since  the  roof  is 
symmetrically  loaded,  the  thrust  at  C  must  be  horizontal,  and 
therefore  the  reaction  which  sujjports  this  end  of  A  C  will  lie 
in  the  line  C  E.  The  centre  of  gravity  of  the  load  on  A  C  is  at 
D,  its  middle  point,  and  the  resultant  of  the  load  will,  if  pro- 
longed upwards,  intersect  C  E  at  E.  Since  the  rafter  is  in 
equilibrium  under  the  load  and  the  reactions  at  C  and  A,  the 
direction  of  the  reaction  of  the  wall  at  A  must  also  pass 
through  E  (compare  Figs.  3  and  4).  Draw  A  E  and  prolong 
ED  to  G.  Let  E  G  be  measured  b}^  such  a  scale  as  to  repre- 
sent the  load  on  A  C.  The  three  forces  meeting  in  the  common 
point  E  will  then  be  equal  to  the  respective  sides  of  the  tri- 
angle AEG,  drawn  parallel  to  them  ;  and,  since  A  G  equals 
E  C,  the  reactions  at  A  and  C  will  be  A  E  and  C  E. 

We  now  decompose  AE  and  CE  into  components  along 
and  transverse  to  the  rafter,  and  have  AF,  direct  compression 
on  the  rafter  at  A,  and  C  F,  direct  compression  at  C.  The 
compression  on  successive  sections  of  the  rafter  increases  from 
C  to  A  by  the  successive  longitudinal  components  of  the  load. 
The  two  components  A  L  and  C  Q,  which,  combined  with  A  F 
and  C  F,  give  the  original  forces  A  E  and  C  E,  are  analogous 
to  the  supporting  forces  of  a  beam  or  truss,  and  through  them 
we  obtain  the  bending  action  of  the  load  on  this  rafter.  If, 
now,  the  rafters  simply  rest  on  the  wall,  being  secured  against 
spreading  by  the  tie  A  B,  the  reaction  A  E  will  be  replaced  by 
the  two  components,  A  I,  the  upward  supporting  force  of  the 
wall,  and  A  G,  the  stress  exerted  by  the  tie  ;  these  two  forces 
give  the  same  stress  and  bending  moments  on  the  rafter  as 
before. 

80.  Comparison  with  Diagram.— Consider,  next,  the 
method  by  diagram.  The  load  is  now  to  be  concentrated  at 
the  joints,  and  in  place  of  E  G,  we  shall  have  A  N  and  C  P, 
each  one-half  of  the  load  on  one  rafter.  Lay  oft"  1-2  to  repre- 
sent the  total  load  on  the  roof,  make  1-3  equal  to  AN  and 
1—4  to  A  I,  and  draw  3-5  and  4-5  parallel  to  the  rafter  and  tie. 


ROOF-TRUSSES.  61 

A  G  will  equal  4-5,  and  therefore  the  stress  in  the  tie  is  given 
correctly  ;  but,  since  A  I— AN  =  AK  =  3-4,  3-5  equals  AD, 
and  this  is  the  stress  given  bj  the  diagram  as  existing  from  A 
to  C,  a  supposition  which  is  true  when  the  load  is  actually 
concentrated  at  the  joints,  but  is  not  true  for  a  distributed 
load.  But  A  D,  or  3-5,  is  equal  to  one-half  of  AF  -)-  F  C,  and 
is  manifestly  the  value  of  the  direct  compression  at  the  middle 
j)oiut  D  of  the  rafter ;  all  of  the  load  from  A  to  D  was,  when 
we  drew  the  diagram,  considered  to  be  concentrated  at  the 
joint  A.  To  3-5,  or  A  D,  we  should  add  D  F,  to  obtain  the 
correct  compression  A  F  at  the  lower  end ;  therefore  a  piece 
which  supports  a  distributed  load  should  have  a  compression, 
equal  to  the  longitudinal  com23onent.of  so  much  of  the  load  as 
is  transferred  to  its  lower  end,  added  to  its  stress  obtained 
from  the  stress  diagram.  The  amount  to  be  added,  however, 
is  generall}^  insignificant  as  compared  with  the  truss  stress. 

The  load  on  the  principal  rafters  of  a  roof-truss  is  usually 
concentrated  at  series  of  equidistant  points,  by  means  of  the 
purlins,  or  short  cross-beams  which  extend  from  one  truss  to 
another,  and  which  are  themselves  weighted  at  a  series  of 
points  by  the  pressure  of  the  second arj^  rafters.  These  second- 
ary rafters,  when  emplojed,  carry  the  boards,  etc.,  and  thus 
have  a  uniformly  distributed  load.  It  is  only  in  cases  where 
purlins  rest  at  other  points  than  the  so-called  joints  that 
bending  action  occurs  in  the  principal  rafters,  or  in  very  light 
trusses  where  the  boards  are  nailed  directly  to  the  main  rafters. 
"We  need  to  determine  the  maximum  bending  moments  on 
such  main  rafters,  on  the  purlins  and  secondary  rafters,  in 
order  to  intelligently  provide  sections  sufficiently  strong  to 
resist  them. 

81.  Bending  Moment,— It  will  first  be  well  to  explain 
what  bending  moment  and  moment  of  resistance  are.  A  horizon- 
tal beam  A  B,  Fig.  32,  supported  at  its  two  ends,  when  loaded 
with  a  series  of  weights,  distributed  in  any  manner,  is  in 
equilibrium  under  the  action  of  vertical  forces,  the  weights 
acting  downwards  and  the  two  supporting  forces  acting  up- 


62  KOOF-TRUSSES. 

wards.  These  supj)ortmg  forces  are  easily  calculated  by  the 
principle  of  the  lever,  or  by  taking  moments  as  explained  in 
§§  26  and  36.  They  will  be  found  graphically  presently.  As 
the  beam  is  at  rest,  there  must  be  no  tendency  to  rotate,  and 
therefore,  if  we  assume  any  point  for  an  axis,  the  sum  of  the 
moments,  that  is  of  the  products  of  each  force  by  its  distance 
from  the  axis,  must  equal  zero.  A  moment  which  tpnds  to 
produce  rotation  in  one  direction  being  called  plus,  one  which 
acts  in  the  other  direction  is  called  minus.  If  then  we  pass 
an  imaginary  vertical  plane  of  section  through  any  point  in 
the  beam,  such  as  E,  the  sum  of  the  moments  on  one  side  of 
the  plane  of  section  must  balance  or  equal  that  on  the  other. 
The  sum  of  these  moments  on  one  side  or  the  other  is  called 
the  bending  moment :  the  reason  for  the  name  will  soon  be 
evident. 

82.  Moment  of  Resistance. — These  bending  moments  on 
o]3posite  sides  of  the  section  in  question  can  balance  one 
another  only  through  the  resistance  of  the  material  of  the 
beam  at  the  section  where  stresses  between  the  particles  are 
set  in  action  to  resist  the  tendency  to  bend.  The  beam 
becomes  slightly  convex,  and  the  particles  or  fibres  on  the 
convex  side  are  extended,  while  those  on  the  concave  side  are 
compressed.  Experiment  shows  that,  for  flexure  within  such 
moderate  limits  as  occur  in  practice,  the  horizontal  forces 
exerted  between  contiguous  particles  vary  uniformly  as  we  go 
from  the  top  of  the  beam  to  the  bottom,  the  compressive 
stress  being  most  intense  on  the  concave  side,  diminishing 
regularly  to  zero  at  some  point  or  horizontal  j^lane,  called  the 
neutral  axis,  then  changing  to  tension  and  increasing  as  we 
approach  the  convex  side.  The  two  sets  of  stresses  reacting 
against  each  other  may  be  represented  to  the  eye  by  the 
arrows  in  the  vertical  section  marked  E'. 

Since  all  of  the  external  forces  are  vertical,  these  internal 
stresses,  being  horizontal,  must  balance  in  themselves,  or  the 
total  tension  must  equal  the  total  compression,  whence  it 
follows  that  the  neutral  axis  must  pass  through  the  centre  of 


EOOF-TRUSSES.  63 

gravity  of  the  section.  To  make  this  fact  clear,  let  one  con- 
sider that  the  distance  of  the  centre  of  gravity  from  any  as- 
sumed axis  or  the  position  of  the  resultant  of  parallel  forces 
is  found  by  multiplying  each  force  or  weight  by  its  distance 
from  that  axis  and  dividing  by  the  sum  of  the  forces.  Now  if 
we  attempt  to  lind  the  centre  of  gravitj'of  a  thin  cross-section 
of  this  beam,  and  take  our  axis  through  the  point  where  the 
centre  of  gravity  happens  to  lie,  the  sum  of  the  moments  of 
the  particles  on  each  side  will  balance  or  be  equal,  and  we  can 
see  that  the  distance  of  each  particle  from  the  axis  will  vary 
exactly  as  these  given  stresses ;  hence  the  neutral  axis  must 
lie  in  the  centre  of  gravity  of  each  cross-section. 

As  these  stresses  are  caused  by  and  resist  the  external  bend- 
ing moment  on  each  side  of  the  section,  the  moment  in 
the  interior  of  the  beam,  made  up  of  the  sum  of  the  products 
of  the  stress  on  each  particle  multiplied  by  its  distance  from 
the  neutral  axis,  or  indeed  from  any  axis,  and  known  as  the 
Tnoment  of  resistance,  must  equal  the  bending  moment  at 
the  given  section.  As  the  tensions  and  compressions  on  one 
side  of  the  plane  of  section  tend  to  produce  rotation  about 
the  neutral  axis  in  the  same  direction,  their  moments  are 
added  together. 

83.  Formula  for  Bending  Moment. — The  bending  mo- 
ment, then,  in  the  beam  AB  of  the  figure,  at  au}-  section  E, 
will  be,  if  Pj  is  the  supporting  force  on  the  right,  W„  W^, 
etc.,  the  weights, 

P2  .  B  E  -  Wi  .  C  E  -  W2  .  D  E  ; 

or,  in  general,  if  L  equal  the  arm  of  any  weight,  and  2  be 
the  sign  of  summation, 

M  (the  bending  moment)  =  P^ .  B  E  —  2  W  .  L, 

it  being  remembered  always  to  take  only  the  weights  between 
one  end  and  the  plane  of  section. 

The  moment  of  resistance,  being  numerically  equal  to  the 
bending  moment,  is  therefore  equal  to  the  above  expression, 
and   the    maximum    stress   at    any   section   can    thence    be 


64  KOOF-TRUSSES. 

determined,  or  the  required  cross-section  to  conform  to  the 
proper  working  stress  for  the  material.  The  weights  on  one 
side  of  the  section  may  all  be  considered  to  be  concentrated 
at  their  common  centre  of  gravity,  or  point  of  application  of 
their  resultant,  so  far  as  the  bending  moment  at  that  section 
is  concerned ;  the  load  when  continuous  is  always  so  taken. 

If  the  reader  will  take  a  special  case,  and,  having  a  beam 
of  known  length  with  weights  in  given  positions,  will  first 
find  the  supporting  forces,  and  then  calculate  the  bending 
moment  on  either  side  of  a  plane  of  section,  he  will  obtain 
the  same  result  with  opposite  signs,  showing  that  the  two 
moments  balance  one  another.  The  numerical  result,  being 
the  product  of  two  quantities,  is  read  as  so  many  foot- 
pounds or  inch-pounds,  according  to  the  units  employed.  As 
the  stress  in  any  material  is  usually  expressed  in  pounds  on 
the  square  inch,  the  latter  units  are  the  better. 

84.  Equilibrium  Polygon. — Let  us  suppose  that  the 
weights  which,  in  Fig.  32,  rest  upon  the  beam  are  transferred 
to  a  cord  at  the  several  points  c,  d,f,  and  g,  vertically  below 
their  former  positions  C,  D,  F,  and  G,  the  cord  itself  being 
attached  to  two  fixed  points  a  and  &,  at  equal  distances  verti- 
cally from  A  and  B.  Let  us  further  supj)ose  that  the  amount 
of  the  weight  at  G  alone  is  at  present  known.  This  cord  can 
be  treated  as  if  it  were  a  frame.  Taking  the  joint  g  into  con- 
sideration, draw  5-4  vertically,  equal  to  the  weight,  then  5-0 
parallel  to  ag  and  4-0  parallel  to  gf.  The  two  lines  just 
drawn  must  be  the  tensions  in  a^  and  gf.  For  the  joint /,/gr 
is  now  known ;  therefore  4-3  parallel  to  the  weight  and  3-0 
parallel  to  fd  will  determine  the  other  forces  at  /.  The 
side  4-3  must  equal  the  weight  at  F,  and  must  lie  in  the  same 
straight  line  with  5-4 ;  for  this  triangle  was  constructed  on 
the  side  4-0  previously  found.  Continuing  the  construction 
for  the  successive  angles  of  the  cord,  we  find  that  a  vertical 
line  5-1  will  represent  by  its  several  portions  the  successive 
weights,  and  that  the  tensions  in  the  diflferent  parts  of  the 
cord  will  be  given  by  the  lines  parallel  to  these  parts,  drawn 


EOOF-TRUSSES.  65 

from  the  points  of  division  of  the  load  line,  and  all  converg- 
ing to  the  common  point  0.  Draw  0-6  horizontally,  and 
hence  parallel  to  a  h ;  this  line  will  be  the  horizontal  com- 
ponent of  the  tension  at  any  point  of  the  cord,  and  is  here 
denoted  by  H.  The  form  assumed  by  the  cord  for  a  given 
distribution  of  weights  is  called  the  Equilibrium  Polygon,  as 
the  system  will  be  in  equilibrium  or  at  rest ;  and  it  is  also 
called  in  mechanics  a  funicular  polygon.  Students  of  mechan- 
ics will  recall  the  fact,  so  easily  shown  here,  that  the  hori- 
zontal component  H  is  a  constant  quantity  at  every  point. 

85,  Reactions. — If  now  the  cord,  instead  of  being  fastened 
to  fixed  points  at  a  and  b,  is  attached  to  the  two  ends  of  a 
rigid  bar  a  b,  and  the  whole  system  is  then  suspended  from  A 
and  B  by  two  short  cords,  its  equilibrium  will  not  be  dis- 
turbed. The  pull  5-0  at  a  will  be  decomposed  into  0-6,  com- 
pression in  ba,  and  6-5,  tension  along  a  A.  Similarly  at 
h,  0-1  will  be  decomposed  into  1-6  along  6B  and  6-0 
along  a  b.  6-0  balances  0-6,  while  1-6  and  6-5  must  be  the 
supporting  forces  at  b  and  a.  As  the  suj)23orting  forces  do 
not  dejDeud  upon  the  form  of  the  frame  or  truss,  the  reac- 
tions which  carry  the  beam  at  B  and  A  must  be  these  same 
quantities. 

86.  Equilibrium  Polygon,  General  Construction. — We 
may  make  the  construction  more  general  by  drawing  an  equi- 
librium polygon  from  any  point  a',  vertically  below  A,  and  find- 
ing the  outline  of  a  cord  which  will  sustain  in  equilibrium  the 
given  weights  at  the  given  horizontal  distances  from  A.  Lay 
off  the  weights  in  succession  from  5  to  1 ;  assume  any  point 
0'  arbitrarily  and  connect  it  with  all  the  points  of  division  of 
the  load  line.  Begin  at  a',  and  draw  a'g'  parallel  to  5-0', 
stopping  at  the  vertical  dropped  from  G;  then  draw  g'f 
parallel  to  4-0',  etc.,  and  finally  c'b'  parallel  to  1-0'.  That 
this  will  be  the  figure  of  a  cord  suspended  from  a'  and  b'  fol- 
lows from  the  preceding  demonstration.  Connect  b'  with  a' ; 
a  line,  parallel  to  b'a',  from  0'  must  strike  the  same  point  6 
which  the   line  from  0,  parallel   to  ha,  touched.     The  sup- 


66  KOOF-TKUSSES. 

porting  forces,  if  h'a'  exists,  will  be  1-6  and  6-5  as  before ; 
but  0'-6'  will  be  the  horizontal  component  H'  for  this  cord. 

87.  The  Equilibrium  Polygon  Gives  Bending  Mo- 
ments.— If  we  turn  again  to  the  first  cord,  attached  at  a  and 
&,  the  piece  a  h  being  dispensed  with,  the  moment  of  all  the 
forces  on  one  side  of  an}^  point,  such  as  e,  must  be  the  bend- 
ing moment  there ;  but  as  the  cord  is  perfectly  flexible  and  at 
rest,  this  bending  moment  will  equal  zero.  Using,  instead  of 
1-0,  its  two  components  1-6  =  P^  and  6-0  =  H,  multiplying 
each  force  by  the  perpendicular  distance  of  its  line  of  action 
from  e,  calling  the  combined  moments  of  the  weights  on  one 
side  of  e  ^  W  .  L  as  before,  and  denoting  the  tendency  to  pro- 
duce rotation  in  opposite  ways  by  opposite  signs,  we  shall 
have,  for  moments  of  forces  on  the  right  of,  and  around  e, 

P2  .  6  A;  -  :2  W.  L  —  H  .  e^  =  0, 
or 

H.  eZ;  =  P2  .  6^-2W.  L. 

But  V,.hh  =  P, .  BE,  and  P, .  BE  -  JSW.L  =  M,  the  bend- 
ing moment  at  the  section  E  of  the  beam,  as  shown  in  §  83  ; 
therefore 

M  =  H  .  eZ;. 

By  a  similar  analysis  of  the  lower  cord  we  have 

Ps  .  ?•  A'  -  S  W  .  L  =  (6-0')  .  e'  Z  =  M. 

From  similarity  of  triangles  le'k'  and  6'0'  6,  we  have 

e'l  :  e'7c'  =  6'-0'  :  6-0', 
or 

(6-00  .e'Z=(6'-0')  .e'k'\ 
therefore 

M=(6'-00  .  e'k'  =  W  .e'k', 

as  in  the  other  case.  The  solution  is  therefore  general,  and 
the  bending  moment  at  any  section  of  the  beam  equals  the 
product  of  H  from  the  stress  diagram  0 1 5  by  the  vertical 
ordinate,  below  the  section,  from  the  cord  to  the  line  connect- 
ing its  two  extremities. 


KOOF-TKUSSES,  67 

88.  Remarks. — The  relative  situations  of  a'  and  h'  will  de- 
pend upon  the  choice  of  the  position  of  0',  and  this  point 
may  be  taken  wherever  convenient.  H'  is  measured  by  the 
same  scale  used  in  plottinji^  5-1,  while  e'li!  must  be  measured 
by  the  scale  to  which  AB  is  laid  oj0f.  The  tAvo  scales,  one 
representing  pounds,  the  other  inches,  need  not  be  numerically 
the  same  ;  their  product  will  be  inch-pounds. 

A  single  load  on  the  beam  will  have  for  its  equilibrium 
polygon  two  straight  lines  from  a'  and  h' ,  meeting  at  a  point 
vertically  under  the  weight.  A  uniformly  distributed  load 
will  give  a  parabola  with  the  maximum  ordinate  at  the  middle 
of  the  span.  This  load  may  be  treated  as  if  concentrated  at 
any  convenient  number  of  points  along  the  beam,  as  we  have 
done  in  getting  the  loads  at  the  several  divisions  of  a  rafter, 
and  the  angles  of  the  polygon  will  lie  in  the  desired  parabola. 
When  the  beam  is  inclined  the  transverse  comj)onents  alone  of 
the  load  produce  any  bending,  as  explained  for  a  uniform 
load  in  §  79.  Wind  pressure  will  act  as  a  uniform  normal  or 
transverse  load  on  the  piece  w^hich  directly  resists  it. 

The  equilibrium  polygon  has  much  more  extended  applica- 
tions in  Parts  II.  and  III. 

89.  Moment  of  Resistance  of  Rectangular  Cross-Sec- 
tion.— Next,  to  determine  the  moment  of  resistance  for  a  par- 
ticular form  of  cross-section  : — Consider  a  beam  of  rectangular 
cross-section,  represented  by  A  B  C  D  of  Fig.  33.  The  inten- 
sity of  stress,  as  shown  at  E',  Fig.  32,  varies  uniforml}'  each 
way  from  the  neutral  axis  which,  lying  through  the  centre  of 
gravity  G  of  the  cross-section,  will  be  at  E  F,  the  middle  of 
the  depth.  The  stress  on  a  square  inch  will  be  most  intense 
on  the  fibres  at  the  edge  A  B  or  C  D,  and  less  intense  on  any 
intermediate  layer,  such  as  I K,  in  the  proportion  of  E I  to 
E  A.  If  then  we  draw  from  G  the  lines  G  A  and  G  B,  and 
imagine  that  the  layer  I K  is  replaced  by  I'  K',  which  has  its 
breadth  diminished  in  the  same  proportion,  the  total  stress 
on  I'  K',  if  of  the  intensity  found  at  A  B,  will  be  equal  to  the 
total  stress  of  less  intensity  actually  existing  on  I K.     The 


68  EOOF-TRUSSES. 

former  stress  will  also  liaise  tlie  same  leverage  about  E  F  as 
does  the  actual  stress  on  I  K.  By  the  same  reasoning  for  all 
layers  of  the  cross-section,  we  obtain  two  triangular,  shaded 
areas,  ABG  and  GDC,  which  may  be  termed  equivalent  areas 
of  uniform  stress  of  intensity  equal  to  the  actual  maximum ; 
one  of  them,  usually  the  upper  one,  when  multij^lied  by  this 
maximum  intensity  of  stress,  represents  the  total  compression, 
and  the  other  the  total  tension  at  the  section.  The  moments 
of  this  tension  and  compression  about  the  neutral  axis  will  be 
most  readily  obtained  by  considering  the  stress,  which  is  now 
uniformly  distributed  over  the  triangle,  as  concentrated  at  its 
centre  of  action,  the  centre  of  gravity  G'  of  the  triangle,  dis- 
tant two-thirds  of  its  height  from  the  apex  G. 

Let  h  represent  the  breadth  and  h  the  height  of  the  cross- 
section  in  inches ;  the  area  of  one  triangle  will  be  ^h  .\h;  and 
the  lever  arm  about  EF  will  be  f .  ^/i.  Let /represent  the 
maximum  stress  on  the  square  inch  at  AB.  Since  the  tension 
and  compression  tend  to  produce  rotation  in  the  same  direc- 
tion, we  add  the  moments  of  the  two  forces  together  and  have 

2  ("2  •/.  ^h\  =  moment  of  resistance  =  ^fhJf. 

Putting  this  value  equal  to  the  bending  moment  M,  we  obtain 

B.'.e'¥=lfbh\ 

If  we  select  the  maximum  value  of  e'k',  introduce  the  safe 
working  stress/ for  the  extreme  fibres,  and  assume  either  &  or 
h,  we  can  compute  the  other  required  dimension,  and  thus 
determine  the  beam  when  of  uniform  section  throughout.  If 
the  cross-section  is  to  vary,  its  moment  of  resistance  at  differ- 
ent points  must  at  least  be  equal  to  the  bending  moments. 
As  the  stiffness  of  the  beam  depends  principally  upon  h,  the 
depth  must  not  be  made  too  small.  If  the  beam  has  too  little 
breadth  the  compressed  edge  will  yield  sideways. 

90.  Moment  of  Resistance  of  T  Section. — It  is  easy  to 
compute  the  size  of  a  beam  of  rectangular  cross-section  by  the 


ROOF-TRUSSES.  69 

above  formula,  but  for  less  regular  sections  the  determination 
of  the  moment  of  resistance  by  this  graphical  method  may 
prove  of  service.  In  applying  it  to  a  beam  of  the  section 
shown  in  Fig.  34  we  must  begin  by  finding  the  centre  of 
gravity  of  the  section.  By  multiplying  each  rectangular  area 
by  the  distance  of  its  centre  of  gravity  from  either  the  top  or 
the  bottom,  adding  these  products,  and  dividing  by  the 
whole  area,  we  find  the  distance  of  the  neutral  axis  from  that 
edge.     If  GI  =  6,  AB  =  &',  GE  =  A,  and  C  A  =  h',  we  have 

r^ — ; — jtt; =  distance  oi  neutral  axis  from  G 1. 

bh-\-  b  h 

The  construction  of  the  shaded  area  A  P  B  needs  no  expla- 
nation, as  it  follows  the  previous  example.  The  stress  on  the 
fibres  at  the  edge  G I  will  not  be  so  great  as  at  the  edge  A  B, 
because  they  are  not  so  far  from  the  neutral  axis.  If  the 
fibres  at  G I  were  removed  to  K  L,  so  as  to  be  equally  remote 
with  AB,  they  would  be  equally  strained.  Then  to  reduce 
the  layer  G I  to  one  which,  if  it  had  the  same  intensity  of 
stress  with  A  B,  would  give  the  same  total  stress  which  now 
exists  on  GI,  project  GI  to  KL,  draw  KP  and  LP,*  and  GT' 
will  be  the  desired  reduced  length.  The  remainder  of  the 
shaded  area  for  the  lower  rectangle  follows  the  usual  rule. 
In  the  same  way,  the  fibres  at  C  D  will  be  projected  at  Q  R, 
and,  by  drawing  Q  P  and  HP,  we  determine  CD',  and  thus 
complete  the  shaded  portion.  These  triangles,  etc.,  can  be 
readily  scaled,  or  computed  from  the  known  proportions  of 
the  beam,  their  centres  of  gravity  found  and  the  moment  of 
resistance  calculated. 

91.  Moment  of  Resistance  of  an  Irregular  Section. — A 
good  example  of  a  section  whose  moment  of  resistance  is  not 
readily  determined  by  computation  alone  is  afi'orded  by  a 
deck-beam.  Fig.  35,  often  employed  in  floors  and  roofs.  It  is 
here  drawn  to  one-quarter  scale,  showing  height  of  section  6 
inches,  breadth  of  flange  A  B  3|  inches,  thickness  of  web  | 
inch,  weight  per  yard  44  lbs. 

*  K  P  and  L  P  should  be  straight  lines,  nearly  touching  C  and  D. 


70  EOOF-TRUSSES. 

The  readiest  way  to  determine  tlie  moment  of  resistance  of 
such  a  cross-section  is  as  follows  : — Transfer  its  outlines  from 
the  book  of  shapes  or  by  such  data  as  you  have  to  a  sheet  of 
heavy  paper,  and  make  a  tracing  for  construction  purposes. 
Cut  the  section  from  the  heavy  paper,  balance  on  a  knife-edge 
and  thus  determine  the  neutral  axis  C  D.  Then  on  the  trac- 
ing draw  K  L  horizontally  at  the  same  distance  from  C  D  that 
S  T  is.  A  B  will  be  projected  at  K  L,  and  lines  from  K  and 
L  to  P,  the  middle  j)oint  of  C  D,  or  the  centre  of  gravity  of 
this  section,  will  cut  AB  at  A' and  B',  making  A'B'  the 
reduced  length  of  A  B,  and  now  considered  to  have  the  same 
stress  per  square  inch  as  exists  at  I  G.  In  the  same  way  the 
end  M  of  M  N  will  be  projected  at  O,  the  point  U  at  Y,  and 
the  lines  from  O  and  V  to  P  will  cut  the  horizontal  lines 
through  M  and  U  at  new  points  in  the  desired  curve.  Thus 
enough  points  are  soon  obtained  to  locate  the  boundary  of 
the  shaded  portion  from  B'  to  P.  The  part  of  the  web  with 
straight  sides  gives  of  course  a  triangle,  found  at  once  by 
drawing  a  line  from  W  to  P.  The  curve  A'  P  corresponds 
with  B'  P.  For  the  lower  portion,  project  E  F  on  T  S,  draw 
lines  to  P,  and  get  in  a  similar  way  enough  points  for  this 
curve.  Cut  out  the  two  shaded  figures  from  the  heavy  paper, 
balance  each  one  over  a  knife-edge  and  thus  determine  their 
respective  centres  of  gravity  Q  and  R.  Calculate  the  area  of 
one ;  the  area  of  the  other  should  exactly  equal  it,  for  the 
total  tension  equals  the  total  compression.  Calling  this  area 
A  and  the  safe  working  stress  on  the  square  inch/,  we  shall 
then  have  for  the  moment  of  resistance 

/.  A.  PQ+/.  A.  PE=/.  A.  QR. 

In  this  example  A  =  1.29  sq.  inches,  P  Q  =  2.12  inches,  and 
PR  =:  2.66  inches.  If  therefore  for  a  static  load  /=  12,000 
lbs.,  the  moment  of  resistance  equals 

12,000  X  1.29  X  4.78  =  74,000  inch-pounds. 

92.  Moment  of  Resistance  of  I  Beam. — In  simpler  cases 
the  required  size  of  beam  to  sustain  a  given  load  is  more  read- 


ROOF-TRUSSES.  71 

ily  found  by  formula.  If  I  beams  are  used,  the  web  being 
thin,  and  the  top  and  bottom  Hanges  alike,  an  approximate 
formula  may  be  used.  If  F  rej)resents  the  area  in  square 
inches  of  the  cross-section  of  either  flange,  W  the  area  of  the 
web,  h  the  dejjth  from  centre  to  centre  of  flanges  or  the  entire 
depth  minus  thickness  of  one  flange  (that  is,  between  centres 
of  gravity  approximately),  and/  the  safe  stress  on  the  square 
inch,  the  moment  of  resistance  is  nearly  equal  to 


CHAPTER  X. 

LOAD   AND   DETAILS. 

93.  Lateral  Bracing.— The  principal  trusses,  if  large, 
should  be  braced  together  in  the  planes  of  the  rafters  to  pre- 
vent wind,  in  a  direction  perpendicular  to  the  gable  ends,  from 
producing  any  lateral  movement.  The  roof  boards,  if  laid 
close,  and  well  nailed,  will  stiffen  trusses  of  moderate  span. 
It  is  often  customary  also  to  fasten  the  trusses  down  to  the 
walls,  especially  in  those  buildings  where  wind  may  get  below 
the  roof.  In  such  cases  it  is  proper  to  consider  and  provide 
for  the  tendency  of  the  wind  to  reverse  the  stresses  in  a  roof 
which  has  a  light  covering. 

94.  Weight  of  Materials. — The  weight  of  the  roof  cover- 
ing can  be  ascertained  in  advance.  The  bending  moments  on 
the  jack-rafters  and  the  purlins  can  then  be  found,  their  sizes 
computed  and  their  weights  added  in.  The  weight  of  the 
truss  must  then  be  assumed  from  such  data  as  may  be  at 
hand.  After  the  diagrams  have  been  drawn  and  the  truss  has 
been  roughly  designed,  its  weight  should  be  calculated  to  see 
how  well  it  agrees  with  the  assumed  weight.  If  this  agree- 
ment is  not  sufficiently  exact,  the  proper  allowance  is  then  to 
be  made. 

Trautwine  says  that,  for  spans  not  exceeding  about  75  feet, 
and  trusses  7  feet  apart,  of  the  type  shown  in  Figs.  11  and  29, 
the  total  load  per  square  foot,  including  the  truss  itself,  pur- 
lins, etc.,  complete,  may  be  taken  as  follows : 

Roof  covered  with  corrugated  iron,  unboarded,      .     .     8  lbs. 

Same  if  plastered  below  the  rafters, 18  " 

Roof  covered  with  corrugated  iron,  on  boards,       .     .  11  " 

72 


EOOF-TEUSSES.  73 

Same  if  plastered  below  the  i*afters, 21  lbs. 

Eoof  covered  with  slate,  unboarded  or  on  laths,    .     .  13  " 

Same  on  boards  li  inches  thick, 16  " 

Same  if  plastered  below  the  rafters, 26  " 

Eoof  covered  with  shingles  on  laths, 10  " 

For  spans  from  75  feet  to  150  feet  it  will  suffice  to  add  4  lbs. 
to  eacli  of  these  totals. 

The  weight  of  an  ordinary  lathed  and  plastered  ceiling  is 
about  10  lbs.  per  square  foot ;  and  that  of  an  ordinary  floor 
of  1-inch  boards,  together  with  the  usual  2  X  12  inch  joists, 
12  inches  apart  from  centre  to  centre,  is  from  9  to  12  lbs.  per 
square  foot.  White  pine  timber,  if  dry,  may  be  considered  to 
weigh  about  25  lbs.,  northern  yellow  pine  35  lbs.,  and  south- 
ern yellow  pine  45  lbs.  per  cubic  foot ;  if  wet,  add  from  20  to 
50  per  cent.  Oak  may  be  reckoned  at  from  40  to  50  lbs.  per 
cubic  foot ;  cast  iron  at  450  lbs.  per  cubic  foot ;  wrought  iron 
at  480  lbs.  per  cubic  foot. 

The  allowance  to  be  made  for  the  weight  of  snow  will 
depend  upon  the  latitude ;  from  12  to  15  lbs.  per  square  foot 
of  roof  will  suffice  for  most  places.  In  some  situations  snow 
may  accumulate  in  considerable  quantities,  becoming  satu- 
rated with  water  and  turning  to  ice ;  but  snow  saturated  with 
water  will  generally  slide  off  from  roofs  of  ordinary  pitch. 
The  weight  of  a  cubic  foot  varies  much ;  freshly  fallen  snow 
may  weigh  from  5  to  12  lbs. ;  snow  and  hail,  sleet  or  ice  may 
weigh  from  30  to  50  lbs.  per  cubic  foot,  but  the  quantity  on 
a  roof  will  usually  be  small. 

95.  Action  of  Materials  under  Stress.— After  the  stresses 
in  the  frame  are  determined,  tlie  several  parts  must  be  designed 
to  withstand  them.  It  is  not  the  purpose  here  to  proportion 
the  members  of  a  truss  and  work  out  the  details.  The  action 
of  materials  under  applied  forces,  the  method  of  calculating 
beams,  ties,  and  struts,  and  the  proper  designing  of  connec- 
tions and  details  are  discussed  at  length  in  the  author's 
"  Structural  Mechanics." 

As  materials,  if  repeatedly  strained  to  an  amount  at  all 
approaching  the  breaking  strain,  will  fail  sooner  or  later,  the 


74 


EOOF-TRUSSES. 


severe  action  weakening  them,  and  as  we  must  provide  for 
unforeseen  and  unknown  defects  of  material  and  workman- 
ship, as  well  as  for  more  or  less  of  shock  and  \ibration,  it  is 
customary  to  so  proportion  the  several  parts  of  a  structure 
that  they  will  be  able  to  resist  without  failure  much  larger 
forces  than  those  obtained  from  the  stress  diagrams.  The 
smaller  the  load  or  stress  on  a  piece  the  greater  number  of 
applications  and  removals  before  the  piece  is  injured  or 
broken.  If  the  stress  is  reduced  so  much  by  increase  of 
cross-section  of  the  member  that  the  j)iece  will  safely  sustain 
an  indefinitely  great  number  of  repetitions  of  it,  such  cross- 
section  will  be  the  proper  one  for  a  piece  in  a  bridge  or 
machine. 

The  stress  arising  from  a  stationary  load,  such  as  the  weight 
of  the  structure,  which  is  constant,  is  not  so  trying  as  repeated 
application  and  release  of  the  same  stress.  The  heavy  wind- 
stresses  determined  in  the  previous  chapters  are  not  likely  to 
occur  more  than  once  or  twice,  if  at  all,  in  the  life  of  the 
structure.  Hence  good  practice  wall  authorize  the  employ- 
ment of  stresses  some  fifty  per  cent,  in  excess  of  those  consid- 
ered allowable  in  first-class  bridge  structures  and  those  sub- 
jected to  frequent  change  of  load,  to  shock  and  vibration. 

96.  Allowable  Stresses. — In  accordance  with  this  view, 
the  following  values  may  be  used,  where  the  wind-pressure  of 
Chapter  IV.  has  been  allowed  for. 


Material. 


White  Oak 

Long-leaf  Southern  Pine. . 

Oregon  Pine  or  Fir 

White  Pine  (Eastern) 

Spruce  

Wrought  Iron.   

"  "    best  quality. 

Soft  Steel , 

Medium  Steel 


Bending 

stress. 

Tension. 

1,600 

1,500 

1,600 

1,400 

1,600 

1,800 

1,400 

800 

1,200 

1,200 

10,000 

12,000 

12,000 

15,000 

14.000 

16,000 

16,000 

18,000 

Compres- 
sion with 
grain. 


1,400 
1,400 
1,300 
1,200 
1,200 


Compres- 
sion across 
grain. 


400 
300 
250 
200 
200 


Compression 
10,000 
12,000 
12,500 
13,750 


Shear  with 
grain. 


180 

150 

200 

100 

100 

Shear 

8,000 

10,000 

10,000 

11,000 


ROOF-TRUSSES.  75 

The  above  values  must  not  be  applied  to  parts  subjected  to 
mo^dng  loads,  such  as  floor-beams  and  suspending  rods  for 
same,  unless  the  load  is  moderate  in  total  amount  and  very 
gradually  ajjplied  and  removed.  For  bridge  work  they  must 
be  reduced  from  '20  to  33  per  cent. 

97.  Tension  Members.— Pieces  in  tension  will  be  liable  to 
break  at  the  smallest  cross-section.  It  is  therefore  economi- 
cal to  enlarge  the  screw-ends  of  long  iron  rods  and  bolts  so 
that  the  cross- section  at  the  bottom  of  the  threads  shall  be 
at  least  as  large  as  at  any  other  point.  It  is  desirable  that 
the  centre  of  resistance  of  the  cross- section  of  struts  and  ties 
shall  coincide  with  the  centre  of  figure,  as  a  deviation  from 
that  j)osition  greatly  weakens  the  piece.  To  calculate  the  net 
or  smallest  cross-section  of  a  tension  member  where  the  pull 
is  axial  or  central  it  is  sufficient  to  divide  the  force  by  the  safe 
working  tensile  stress.  Allowance  must  be  made  for  diminu- 
tion of  cross-section  by  any  cutting  away,  bolt  or  rivet  holes. 

98.  Compression  Members. — For  very  short  pieces  or 
blocks  in  compression,  whose  lengths  do  not  exceed  six  times 
the  least  dimension,  the  same  process  may  be  followed.  But 
as  the  length  increases  the  strut  has  a  tendency  to  yield 
sideways  when  compressed,  and  the  cross-section  must  be 
increased.  Let  I  be  the  length  of  the  strut  in  inches,  h  ita 
least  external  diameter  in  inches,  and  r  the  least  radius  of  gyra- 
tion of  its  cross-section  in  inches.  Then  the  safe  mean  work- 
ing compressive  stress,  to  be  used  as  a  divisor  of  the  given 
force,  to  find  the  cross-section  of  the  strut,  will  be,  for  piecee 
with  flat,  securely  bedded  ends,  or  ends  fixed  in  direction  by 
bolting  or  riveting. 

Southern  Pine 1200  -  12-. 

A 

White  Pine 1000  -  loj-. 

Soft  Steel 12500  -  42-. 

r 

Medium  Steel 13750  -  48-. 

r 


76  ROOF-TRUSSES. 

If  the  struts  are  jointed  at  their  ends  bj  pin  connections, 
or  are  so  narrow  as  to  readily  yield  sideways  at  these  points^ 
double  the  subtractive  term  in  the  preceding  formulas. 

The  hand-books  issued  by  the  steel  manufacturers  give  the 
sections  and  weights  of  the  various  rolled  shapes,  the  values 
of  r  for  different  axes,  the  safe  loads  for  beams  of  different 
spans,  details  of  construction,  and  miscellaneous  useful  infor- 
mation. The  inexperienced  designer  should  exercise  great 
care  in  computing  compression  members,  and  be  sure  that  the 
least  radius  of  gyration  is  used  in  the  formula. 

Pieces  subjected  alternately  to  tension  and  compression 
should  have  a  materially  larger  section  than  would  be  required 
for  either  stress  alone. 

Cast  iron  is  not  in  favor  with  the  best  designers  for  any 

but  short  compression  pieces,  packing  blocks  and  pedestals, 

although  it  is  still  employed  for  columns.     The  formula  for 

I 
cast  iron  may  be  15,000  —  50-. 

99.  Beams. — The  values  of  /  to  be  used  in  the  moment  of 
resistance,  for  pieces  subjected  to  bending,  are  marked  bend- 
ing stress  in  the  preceding  table.  In  determining  the  moment 
of  resistance  of  a  piece  exposed  to  bending,  or  in  calculating- 
the  cross-section  required  at  the  point  of  maximum  bending^ 
moment,  allowance  must  be  made  for  portions  cut  away  on. 
the  tension  side  in  attaching  fastenings,  bolting  or  riveting 
together  parts,  and  also  on  the  compression  side  unless  the 
holes,  etc.,  are  so  tightly  filled  that  the  compression  can  be 
fairly  considered  as  resisted  by  those  portions  also. 

Those  pieces  which  resist  both  a  bending  moment  and  a 
direct  stress  may  first  be  designed  to  safely  carry  the  bending 
moment,  and  then  the  dimension  transverse  to  that  in  which 
the  piece  will  bend  may  be  so  much  increased  that  the  added 
slice  will  resist  the  direct  pull  or  thrust.  If  that  force  is 
thrust,  it  will  be  well  to  test  the  size  of  the  piece  by  the  for- 
mula on  the  preceding  page. 

100.  Pins  and  Eyes. — A  reasonable  rule  for  proportioning; 


ROOF-TRUSSES.  77 

pins  and  eyes  of  tension  bars  is  as  follows : — Make  the  diam- 
eter of  the  pin  from  three-fourths  to  four-fifths  of  the  width 
of  the  bar  in  flats,  and  one  and  one-fourth  times  the  diameter 
of  the  bar  iu  rounds,  giving  the  eye  a  sectional  area  of  fifty 
per  cent,  in  excess  of  that  of  the  bar.  The  thickness  of  flat 
bars  should  be  at  least  one-fourth  of  the  width  in  order  to 
secure  a  good  bearing  surface  on  the  pin,  and  the  metal  at  the 
eyes  should  be  as  thick  as  the  bars.  As  tlie  bending  moment 
on  a  pin  generally  determines  its  diameter,  pieces  assembled 
on  a  pin  should  be  packed  closely,  and  thojs:;  having  ojiposing 
stresses  should  be  brought  into  juxtaposition  if  possible. 

101.  Details. — Very  close  attention  must  be  given  to  all 
minor  details ;  to  so  proportion  all  the  parts  of  a  joint  that  it 
will  be  no  more  likely  to  yield  in  one  way  than  another ;  to 
■weaken  as  little  as  possible  the  pieces  connected  at  a  splice  ; 
to  give  suflicient  bearing  surface  so  as  to  bring  the  intensity 
of  the  comj)ression  on  the  surface  within  proper  limits :  to 
distribute  rivets  and  bolts  so  as  to  give  the  greatest  resist- 
ance with  the  least  cutting  away  of  other  parts ;  to  keep  the 
action  line  of  every  piece  as  near  its  axis  as  possible  ;  and  to 
examine  all  sections  and  parts  for  tension,  compression,  and 
shear.  The  failure  of  a  joint  or  connection  is  as  fatal  to  a 
frame  as  to  have  a  member  too  small  for  the  stress  upon  it. 

The  following  sections  are  quoted  from  the  author's  "  Struc- 
tural Mechauics  " : 

102.  Framing  of  Timler:  Splices.  —  Sketches  XL  to 
XVI.  in  Plate  IV.  represent  diflereut  methods  of  splicing  a 
timber  tie.  In  each  case  the  smallest  cross-section  of  the 
timber  determines  the  amount  of  tension  that  can  be  trans- 
mitted. The  shoulders  are  in  compression,  and  the  longitu- 
dinal planes  between  the  shoulders  are  in  shear.  In  XI.,  for 
equal  strength,  the  depth  of  the  two  opposite  shoulders  or 
indents  should  be  to  the  remaining  depth  of  the  timber  as  the 
safe  unit  tensile  stress  is  to  the  safe  unit  compression  along 
the  grain.  The  shearing  length,  on  either  timber  or  clamp, 
should  be  to  the  depth  of  shoulder  as  the  safe  unit  compres- 


78  ROOr-TKlTSSES. 

sion  is  to  the  safe  unit  sliear.  In  actual  practice,  unless  con- 
siderable dependence  is  placed  upon  the  resistance  of  the 
bolts  against  shearing  through  the  timber,  the  splice  should 
be  much  longer  than  shown.  If  the  two  clamps  are  of  stronger 
wood  than  the  main  timber,  they  need  not  together  have  so 
much  depth  as  the  net  depth  of  the  timber.  The  iron  strap 
in  XIY.  illustrates  the  same  principle.  The  bolts  are  usually- 
small,  and  serve  mainly  to  balance  the  moment  set  up  on  each 
clamp  by  the  pressure  on  the  shoulder  and  the  tension  in  the 
neck.  The  modification  in  XII.  permits  the  introduction  of  the 
bolts  without  reducing  the  net  section  of  the  timber.  In  XIIL, 
each  indent  is  only  half  the  previous  depth,  with  obvious 
economy  of  the  main  timber,  and  increase  of  shearing  area  of 
clamp  and  timber  without  lengthening  the  clamps.  It  is  much 
more  difiicult  to  fashion,  however,  and  it  is  not  probable  that 
both  shoulders  on  one  half  will  bear  equally. 

XV.  and  XVI.  are  scarfed  joints.  The  tension  sections, 
the  compression  shoulders  and  the  longitudinal  shearing 
planes  should  again  be  properly  proportioned  here.  In  XV., 
but  one-third  of  the  timber  is  available,  if  unit  tension  and 
compression  have  the  same  numerical  value,  while  in  XVI. 
one-half  of  the  stick  is  useful;  but  the  latter  joint  is  more 
troublesome  to  fashion.  The  bolts  serve  to  resist  the  moment 
which  tends  to  open  the  joint,  and,  by  resisting  it,  cause  a 
fairly  uniform  distribution  of  stress  in  the  critical  section. 
The  bolt-holes  do  not  weaken  the  timber.  Sometimes  the  ex- 
treme ends  of  the  scarf  are  undercut  to  check  the  tendency  to 
spring  out  when  the  bolts  are  not  used.  Keys  may  be  driven 
through  places  cut  for  them  at  the  shoulders.  The  joint  can 
then  be  readily  assembled  and  forced  to  place.  These  sketches 
show  that  timber,  although  possessing  good  tensile  strength, 
is  ill-adapted  for  ties,  on  account  of  the  great  loss  of  section 
in  connections  and  joints. 

103.  Struts  and  Ties. — The  connection  of  a  strut  and 
tie  in  wood  is  illustrated  in  II.,  III.,  IV.  and  VII.  The  shrink- 
ago  of  the  pieces  of  II.  in  seasoning  tends  to  open  botli  por- 


EOOF-TRUSSES.  79 

tions  of  the  joint  by  changing  the  angles  ;  but  the  bearing  of 
the  strut  is  still  central,  if  only  on  a  small  area.  The  com- 
pression of  the  tie  across  the  grain  may  be  large  in  such  a 
case,  and  the  introduction  of  a  block,  as  in  IV.,  will  remedy 
such  a  difficulty  as  well  as  that  from  shrinkage.  The  block 
below  is  the  wall-plate,  for  distributing  the  truss  load  along 
the  wall.     It  is  subjected  to  compression  across  the  grain. 

If  the  shearing  area  to  the  left  in  these  four  cases  is  not  suf- 
ficient, the  bolt  or  strap  is  a  wise  provision  to  take  up  the 
horizontal  component.  The  bolt,  if  a  little  oblique  to  the  strut, 
as  shown,  holds  at  once  by  tension,  to  some  degree,  and  not 
alone  by  shear.  It  also  relieves  the  smallest  section  of  the  tie 
from  a  part  of  the  tension.  The  square  shoulders  of  III.  are 
good,  if  the  timber  is  seasoned,  as  the  bearing  is  then  over  the 
whole  end  of  the  strut,  and  the  tie  is  not  weakened  any  more 
than  in  II.,  while  the  joint  is  more  simply  laid  out.  The  strap 
of  VII.  gives  a  satisfactory  bearing  for  the  strut,  but  the  fast- 
enings of  such  a  strap  are  often  weaker  than  the  strap  itself. 
The  holes  in  it  may  well  be  enlarged  hot,  without  removal  of 
metal  and  diminution  of  cross-section. 

In  VIII.,  IX.  and  X.  are  shown  connections  of  struts  which 
may  at  some  time  be  called  on  to  resist  tension,  or  which  may 
be  relieved  of  stress  and  become  loose.  The  tenon  in  VIII. 
must  be  pinned  to  carry  tension;  and  the  pin  will  resist  but 
little  before  shearing  out  of  the  tenon  or  splitting  oflf  the  side 
of  the  other  timber  by  tension  across  the  grain.  The  tenon 
should  be  fashioned  as  indicated,  with  sufficient  area  at  the 
left-hand  edge  to  carry  the  perpendicular  component  of  the 
thrust  of  the  strut  as  compression  across  the  grain,  and  suf- 
ficient cross-section  not  to  shear  off.  The  size  of  the  strut 
must  be  determined,  not  only  by  the  column  strength,  but  by 
the  area  necessary  to  prevent  crushing  the  piece  against  which 
it  abuts.  This  remark  applies  to  IX.  and  X.  also.  The  abil- 
ity of  IX.  to  carry  tension  depends  on  the  resistance  of  the 
nut,  which  is  slipped  into  a  hole  at  the  side,  to  shearing  out 
along  the  strut,  or  crushing  the  fibres  on  which  it  bears,  the 


80  ROOF-TRUSSES. 

latter  method  of  failure  being  the  more  likely,  unless  the  nut 
is  quite  near  the  end  of  the  strut.  The  strap  on  X.  is  very 
effective,  and  the  arrangement,  if  inverted,  will  serve  as  a  sus- 
pending piece,  although  a  rod  is  better.  Many  of  these  con- 
nections are  serviceable  in  other  positions. 

To  keep  a  strut  from  crushing  the  side  of  a  timber,  a  con- 
nection may  be  employed,  as  in  the  lower  part  of  I.  This 
device  may  be  economical,  if  a  number  of  such  joints  are  to 
be  made,  and  it  is  superior  to  a  mortise  in  work  exposed  to 
the  weather,  as  there  is  no  place  for  water  to  lodge.  The  post 
in  XVIII.  is  capped  by  a  similar  device  for  distributing  and 
thus  reducing  the  unit  pressure  on  the  other  piece.  Lateral 
displacement  is  provided  against  in  both  cases  by  ribs  on  the 
castings. 

Strut  connections  are  shown  in  XIX.  and  XX.,  with  a  tie- 
rod  in  addition.  The  broad,  flat  washer  reduces  the  unit  com- 
pressive stress  on  the  wood  under  it :  the  lip  keeps  water  out 
of  the  joint.  Shrinkage  and  a  slight  deflection  of  the  frame 
under  a  load  will  cause  the  mitre  joint  in  XIX  to  bear  at  the 
top  only,  throwing  the  resultant  stress  out  of  the  axis  of  the 
respective  compression  members  and  causing  the  unit  com. 
pression  at  top  edge  of  the  joint  to  be  very  high.  The  joint 
in  XX.  gives  a  better  centre  pressure,  and  is  easily  made ; 
the  upper  piece  is  simply  notched  for  one-half  its  depth,  and 
the  upper  and  lower  edges  come  on  the  mitre  line  of  XIX. 
The  connection  of  XX  .,  by  the  insertion  of  an  iron  plate  or 
a  block  of  wood,  secures  a  certain  continuity  or  rigidity  in 
the  joint,  to  resist  a  moderate  amount  of  bending  moment. 
The  two  pieces  might  have  been  halved  together.  XXYI.  is 
like  VIII.,  without  provision  for  tension,  which  is  usually  un- 
necessary. The  roof  purlin  with  its  block  is  also  shown  in 
relative  position. 

104.  Beam  Connections. — In  I.  and  XVIII.  are  shown 
supports  of  beams  on  posts.  The  double  or  split  cap  of  I.  is 
serviceable  where  several  posts  are  to  be  connected  laterally, 
as  in  a  trestle  bent,  and  it  is  desired  to  do  away  with  mortises. 


ROOF-TRUSSES.  81 

Bolts  sliould  be  put  transversely  through  the  caps  and  top 
of  the  post.  A  comparatively  wide  bearing  for  the  beam, 
without  the  use  of  large  timber  caps,  may  be  here  secured. 
Lateral  bracing,  as  in  XXVIII.,  will  be  needed.  An  indirect 
and  intermediate  support  for  a  beam,  by  two  inclined  braces, 
is  seen  in  XXV.,  and  the  reverse  case  is  represented  in  XXVII. 
A  mortise  and  tenon  of  usual  proportions  are  shown  in  XVII. 
The  ordinary  wall  bearing  for  joists  may  be  seen  in  the  lower 
left-hand  corner.  The  slanting  end  is  a  wise  provision  to  pre- 
vent harmful  action  of  the  loaded  joist  on  the  wall,  and  it  pro- 
motes ventilation  of  the  timber. 

The  usual  way  of  connecting  two  floor  joists  or  beams, 
when  their  upper  surfaces  are  to  be  at  one  level,  is  drawn  in 
VI.  The  nearer  the  mortises  are  to  the  neutral  axis,  the  less 
the  weakening  of  the  pieces  in  which  they  are  cut ;  on  the 
other  hand,  the  farther  the  two  tenons  are  aj^art,  the  more 
firmly  is  the  tenoned  joist  held  against  lateral  twist.  The 
shouldered  tenon,  indicated  by  the  dotted  lines  at  the  left,  is 
designed  to  attain  both  objects,  to  weaken  the  mortised  piece 
as  little  as  possible  and  to  have  a  considerable  depth  of  tenon, 
as  well  as  a  long  tongue  projecting  entirely  through.  The 
work  of  framing  is  considerably  more  than  in  the  former 
case. 

105.  Wooden  Built  Beams. — If  seasoned  material  is  at 
hand,  and  large  timbers  are  too  expensive,  a  useful  beam 
may  be  built  up  by  placing  planks,  from  two  to  four  inches 
thick,  edge  to  edge,  and  then  thoroughly  nailing  or  spiking 
boards  on  both  sides  at  an  angle  of  45°  with  the  length  of  the 
beam,  and  sloping  in  opposite  directions  on  the  two  sides.  By 
due  regard  to  jointing  and  nailing  a  beam  of  considerable  span 
may  be  made  at  moderate  cost.  The  construction  can  be 
doubled  if  necessary. 

Another  compound  beam  is  seen  m  XXV.  The  keys  and 
bolts  resist  the  shear  along  the  neutral  axis ;  the  horizontal 
sticks  are  butted  together  on  the  compression  side,  and  are 
strapped  by  the  metal  clamp  indicated  to  carry  tension,  if 


82  KOOF-TRUSSES. 

necessary.     The  small  block  behind  the  clamp  keeps  it  in 
place. 

106.  Curved  Beams. — Planks  placed  side  by  side,  as  in 
XXII.,  cut  to  the  form  of  a  curved  beam  or  arched  rib,  and 
bolted  together  to  prevent  individual  lateral  yielding,  are  quite 
effective,  if  the  grain  of  the  wood  does  not  cross  the  curve  too 
obliquely.  Hence,  when  the  curvature  is  considerable,  it  may 
be  advisable  to  use  short  lengths,  which  must  break  joint  in 
the  several  parallel  pieces.  It  is  well  to  make  a  deduction  of 
one  piece  in  computing  the  strength  of  the  member  at  any 
section.  The  ratio  of  strength  of  this  combination,  when  well 
bolted  together,  to  that  of  a  solid  stick  may  be  considered  to 
be  as  71  —  1  to  n,  where  n  is  the  number  of  layers. 

If  the  planks  are  bent  to  the  curve  and  laid  upon  one 
another,  as  in  XXIII,  this  combination  is  not  nearly  so  effect- 
ive as  the  former,  but  it  can  be  more  cheaply  made.  The 
lack  of  efficiency  arises  from  the  unsatisfactory  resistance 
offered  to  shear  between  the  layers  by  the  bolts  or  spikes. 
The  strength  to  resist  bending  moment  will  be  intermediate 
between  that  of  a  solid  timber  and  that  of  the  several  planks 
of  which  it  is  composed,  with  a  deduction  of  one  for  a  prob- 
able joint. 

If  the  curved  member  has  a  direct  force  acting  upon  it 
and  a  moment  arising  from  its  curvature,  the  treatment  will 
follow  the  same  lines;  but  the  joints,  if  there  are  any,  will  be 
more  detrimental  in  case  there  is  tension  at  any  section. 
Such  curved  pieces  are  sometimes  used  in  open  timber  trusses 
for  effect,  but  their  efficiency  is  low  on  account  of  the  large 
moment  due  to  the  curvature.     XXIL  is  the  stiffer. 

The  joints  and  connecting  parts  in  all  timber  construction 
should  be  jjroportioned  in  detail  for  such  tension,  compression 
and  shear  as  they  may  have  to  withstand.  Often  the  three 
kinds  of  stress  occur  in  different  parts  of  one  joint  or  connec- 
tion. 

107.  Iron  Roof-truss.  —  Joints  I.  to  IV.,  Plate  V., 
represent  ways  of  connecting  the  several  pieces  of  a  compara- 


ROOF-TUUSSES.  83 

tively  liglit  roof-truss.  All  the  members  are  made  with  angles, 
and  at  several  points  both  legs  of  the  tension  angles  are  fast- 
ened. Joint  I.  comes  between  II.  and  III.,  and  IV.  comes 
perpendicularly  opposite  it.  The  number  of  rivets  in  each  of 
the  ties  and  centre  member  of  II.  depends  upon  the  force  in 
the  particular  piece  and  the  rivet  shearing  value  and  bearing 
value  in  the  thinnest  piece.  The  number  of  rivets  in  the  raf- 
ter  likewise  depends  upon  the  force  it  carries,  unless  the  two 
rafters  are  supposed  to  abut  and  to  transmit  so  much  of  the 
horizontal  component  as  does  not  come  through  the  inclined 
ties,  a  treatment  not  to  be  commended.  The  two  angle-irons 
of  the  rafter,  being  in  compression,  should  be  connected  at 
intervals  by  a  rivet  and  filling  piece  or  thimble.  The  number 
of  rivets  through  the  rafter  and  connection  plate  at  I.  need  only 
be  enough  to  transmit  the  force  from  one  diagonal  to  the  raf- 
ter. Study  the  necessity  for  rivets,  and  do  not  add  all  the 
rivets  in  abutting  pieces  to  obtain  the  number  in  a  main 
member. 

Similarly,  in  IV.,  the  first  four  or  possibly  five  rivets  on 
the  left  in  the  horizontal  member  balance  the  rivets  in  the 
inclined  tie  on  the  right ;  the  six  remaining  rivets  seen  and 
three  others  unseen,  on  the  left  of  the  splice,  balance  the  same 
number  in  the  smaller  angle.  Note  how,  by  an  extension  of 
the  connecting  plate  and  a  short  plate  below,  the  main  tie  is 
neatly  spliced  and  reduced  in  section. 

The  rafter  at  III.  has  more  rivets  than  at  the  upper  end 
because  the  thrust  is  somewhat  greater.  The  rivets  in  the  tie 
at  that  connection  will  practically  equal  those  at  the  other  end 
of  the  same  piece.  The  black  holes  at  VI.  indicate  the  rivets 
to  be  inserted  at  the  time  of  erection,  and  these  should,  in  good 
practice,  exceed  the  number  called  for  in  joints  riveted  in  the 
shop.  They  must  carry  the  load  and  resist  the  moment  of  the 
horizontal  component  due  to  the  wind  pressure,  which  passes 
down  the  post  IX.  as  shear.  The  post  is  subjected  to  bend- 
ing moment  as  well  as  compression,  and  hence  has  one  dimen- 
sion much  greater  than  the  other.     Bracing  perpendicular  to 


84  ROOF-TRUSSES. 

the  plane  of  the  truss  is  needed  to  resist  wind  pressure  on  the 
end  of  the  structure.  Columns  and  comjDression  members,  in 
structural  work  of  any  kind,  if  joined  one  to  another,  must  be 
thoroughly  stayed  against  lateral  movement. 

Pin-connected  roof-trusses  resemble  in  their  details  the 
joints  of  the  next  section. 

108.  Pin-connected  Bridge. — Ordinary  details  in  a  pin- 
jointed  bridge  truss  of  moderate  span  are  shown  in  VII.,  VIII. 
and  XVI.  The  position  of  the  splice  in  the  top  chord  is  near 
the  pin.  The  splice-plate  may  be  extended  to  reinforce  the 
pinhole,  if  required.  The  ends  of  the  chord  pieces  are 
machined  plane  and  parallel,  and  only  enough  rivets  are  then 
used  in  the  splice  to  insure  the  alignment.  The  pin  is  usually 
placed  in  the  centre  of  gravity  of  the  chord  section.  The 
connection  plates  are  seen  below,  to  keep  the  sides  of  the 
chord  from  spreading;  the  rest  of  the  panel  length  is  usually 
laced.  Another  chord  section,  employing  channels,  is  drawn 
at  XI. 

XII.,  XIII.  and  XIV.  show  sections  for  posts.  They  offer 
facilities  for  the  central  support  of  floor-beams.  Post  flanges 
are  sometimes  turned  out,  sometimes  in.  The  floor-beam,  of 
plate-girder  type,  is  riveted  at  XVI.  to  the  post  through  the 
holes  shown.  This  attachment  stiffens  the  trusses  laterally 
and  is  much  superior  to  hangers.  Top  and  bottom  lateral 
bracing,  to  convey  the  wind  pressure  to  the  abutments,  is 
needed  in  the  planes  of  the  chords,  and  portal  bracing  at 
each  end  to  throw  the  wind  pressure  from  the  top  system  into 
the  end  posts,  which  convey  it  to  the  abutments  as  shear, 
with  the  accompanying  bending  moments  in  those  posts. 

The  posts  go  inside  of  the  top  chord,  as  do  the  main 
diagonals  or  ties,  which  come  next  to  the  posts.  The  bottom 
chord  bars  are  on  the  outside,  one  of  those  running  towards 
the  middle  of  the  span  being  usually  the  farthest  out. 

109.  Riveted  Bridges. — A  riveted  Warren  girder  or 
latticed  truss  is  shown  below.  These  details  are  not  for  con- 
secutive joints.     The  increase  of  chord  section,  when  neces- 


ROOF-TRUSSES.  86 

sary,  is  indicated  at  XIX.  If  the  truss  is  loaded  on  the  top, 
interior  diagonal  bracing,  drawn  at  XXI.,  must  be  used. 
When  the  truss  is  a  lattice,  the  web  members  are  connected 
at  intersections  to  stiffen  the  compression  members,  as  at 
XX.,  or  preferably  as  at  V,  or  at  XV.,  if  the  web  is  double. 
Horizontal  lateral  bracing  must  not  be  overlooked. 

X.   is  one  form   of  section  of  a  solid  bridge  floor.     Beet- 
angular  sections  are  also  used. 


\ 


ROOF  TRUSSES 


ROOF  TRUSSES, 


ROOF  TRUSSES 


<^ 

■■-yf  ^ 

^ -- 

ROOF  TRUSSES. 


INDEX. 


PAGE 

Action  of  wind  2i 

Allowable  stresses 74 

Analysis,  order  of 5 

Beams,  designing 76 

Bending  moment 61 

"  "        formula 63 

"             "        from    equilibri- 
um polygon 66 

Bending  moment  on  rafter 59 

Bracket  truss 12 

Cambering  lower  tie,  effect  of . . .  14 

Change  of  diagonal 18,  42 

Compression   and  tension,  to  dis- 
tinguish between 6,  32 

Compression  members,  designing  75 

Curb  roof,  truss  for 33 

"       "      without  roller 34 

"       "      with  roller 36 

Curved  members 46 

Curved  roof-truss 37 

Details 77 

Diagonal,  change  of 18,  42 

' '  reversal  of 53 

Diagonals  in  same  quadrilateral, 

two 18 

Distribution  of  load 15 

Equilibrium  polygon 64  to  67 

Eyebars 76 

Example,  general 56 

Flat  roofs,  trusses  for 16 

Fink  truss 13.  54 

Forces  not  applied  to  joints — 50,  59 

Hammer-beam  truss 46,  55 

"           "        "        amount    of 
horizontal  thrust 47 


PAGE 

Horizontal  thrust  .ndeterminate. .  48 

"  "       trusses  with 44 

Howe  truss 20 

Imaginary  f  Drees,  use  of 50,  58 

Inclined  forces,  trusses  under.  .22,  33 

' '         reactions 7,  8 

Irregular  section,  moment  of  re- 
sistance of 69 

I  section,  moment  of  resistance,  70 

Joints,  loads  between. 50.  59 

"        loads  on  all 14,19 

King-post  truss 9 

Lateral  bracing 72 

Load  and  details 72 

"      between  joints 50.  59 

"      on  all  joints 14,  19 

Lower  tie,  effect  of  cambering 14 

Materials  under  stress,  action  of,  73 

"  weight  of 72 

Method  of  trial  and  error 55 

Moment,  bending 61 

Moment,  formula  for  bending.  ...  63 
from  equilibrium    poly- 
gon       66 

Moment  of  resistance 62 

"        "        "         irregular  sec- 
tion   69 

Moment  of  resistance,  I  section.  . .   70 
"        "        "  rectangular 

section 67 

Moment  of  resistance,  T  section. ..  68 
Moments,  reactions  found  by.  .17,  21 

Moving  load 21 

Notation 2 

Order  of  analysis 5 

87 


88 


INDEX. 


PAGE 

Pins 77 

Pulonceau  truss 13,  54 

Polygou,  equilibrium 64,  Ho 

Pratt  truss 20 

Pressure,  wind- 23 

Principle  of  reciprocity 6 

Queen-post  truss  16 

Kafier,  bending  moment  on 59 

Railroad-station  roof 53,  56 

KeactioQS     found    by    moments, 

17,  21,  24 

Reactions  from  wind 24,  28.  35 

Reciprocity,  principle  of 6 

Rectangle,  moment  of  resistance,  67 

Resistance,  moment  of 6"i 

"  "        "  for  various 

sections 67  to  70 

Reversal  of  diagonal 53 

Roller  bearing,  effect  of 27,  36 

Roof,  truss  to  conform  to 19 

Roof -truss,  wooden 10 

Scissor  truss 44 

Snow  diagram 33 

Snow,  weight  of 73 

Special  solutions 53 

Stress,  action  of  materials  under. .  73 

"       determining  kind  of 6,  32 

Stresses,  allowable  74 

"        in  triangular  frame 3,  4 

Superfluous  pieces 11 

Tension  and  compression,  to  dis- 
tinguish between 6,  32 

Tension  members,  designing 75 

Three  forces  unknown 13,  53 

Trapezoidal  truss,  equal  loads. ...  16 
"  "     unequal    loads,  16 

Trial  and  error,  method  of 55 

Triangle  of  forces 1 

"         "  external  forces 2,  4 


PAGE 

Triangular  truss. .. .  7 

Truss    conforming    to    shape    of 

equilibrium  polygon 16,  39 

Truss  for  curb  or  mansard  roof. ..  33 

Truss,  Fink 13,  54 

Hammer-beam 46,  55 

Howe 20 

"      King-post 9 

Pratt 20 

"      Polonceau 13,54 

"      Queen-post 16 

"      Scissor 44 

"       Trapezoidal 16 

' '      Warren 19 

Truss  to  conform  to  roof 19 

"     with  roller  bearing 27,  ::6 

Trusses  for  flat  roofs 16 

"   halls 18 

"       under  vertical  forces 7,  16 

"      inclined     "     ...22,33 

"      with  horizontal  thrust 44 

T  section,  moment  of  resistance,  68 
Use  of  two  diagonals  in  quadri- 
lateral    18,  40 

Vertical  forces,  trusses  under..?,  16 

Warren  girder 19 

Weight  of  materials 72 

Wind,  action  of 22 

"     diagram,  reactions,  24,  28.  29,  41 
stresses,  25,  30,  34,  43 
"     on  alternate  sides,  change  of 

stress 26,  31,  43 

Wind-pressure 23 

"            "        on  curb  or  man- 
sard roofs 33 

Wind-pressure  on  curved  roofs..   37 
"            "          "  pitched  or  ga- 
ble roofs 23 

Wooden  roof -truss 10 


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Part  I. — Propagation,  Culture,  and  Improvement i2mo,  i  50 

Part  II. — Systematic  Pomology izmo,  i  50 

Downing's  Fruits  and  Fruit-trees  of  America 8vo,  5  oo- 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  so- 
Practical  Farm  Drainage lamo,  i   o 

Green's  Principles  of  American  Forestry.     (Shortly.) 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo,  2  o 

Kemp's  Landscape  Gardening i2mo,  2  .so 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration i2mo,  i  5 

Sanderson's  Insects  Injurious  to  Staple  Crops i2mo,  i  5 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 
Insects  Injuring  Fruits.     (In  preparation.) 

Stockbridge's  Rocks  and  Soils 8vo,  2 

Woll's  Handbook  for  Farmers  and  Dairymen i6mo,  i  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo- 

Birkmire's  Planning  and  Construction  of  American  Theatres 8vo,  3  00 

Architectural  Iron  and  Steel 8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  00 

Planning  and  Construction  of  High  Office  Buildings 8vo,  3  50 

Skeleton  Construction  in  Buildings 8vo,  3  00 

Briggs's  Modern  American  School  Buildings 8vo,  4  00 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  4  00 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

Fireproofing  of  Steel  Buildings 8vo,  2  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  00 

Theatre  Fires  and  Panics ....    i2mo,  i   50- 

i 


Hatfield's  American  House  Carpenter 8vo,  5  00 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo,  75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  00 

Kidder's  Architect's  and  Builder's  Pocket-book i6mo,  morocco,  4  00 

Merrill's  Stones  for  Building  and  Decoration Svo,  5  00 

Monckton's  Stair-building 4to,  4  00 

Patton's  Practical  Treatise  on  Foundations Svo,  5  00 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry Svo,  i  50 

Snow's  Principal  Species  of  Wood Svo,  3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 
(Shortly.) 

Wait's  Engineering 'and  Architectural  Jurisprudence Svo,  6  00 

Sheep,  6  50 
Law  of  Operations  PreUminary  to  Construction  in  Engineering  and  Archi- 
tecture   _ . .  Svo,  5  00 

Sheep,  5  50 

Law  of  Contracts Svo,  300 

Woodbury's  Fire  Protection  of  Mills Svo,  2  50 

Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Suggestions^for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

i2mo,  I   25 

The  World's  Columbian  Exposition  of  1893 Large  4to,  i  00 


ARMY  AND,  NAVY. 

Bemadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Bruff's  Text-book  Ordnance  and  Gunnery Svo,  6  00 

Chase's  Screw  Propellers  and  Marine  Propulsion Svo,  3  00 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squire's  Polarizing  Photo-chronograph Svo,  3  00 

Cronkhite's  Gunnery  for  Non-commissioned  Officers 24mo.  morocco,  2  00 

*  Davis's  Elements  of  Law Svo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States Svo, 

*  Sheep 

De  Brack's  Cavalry  Outpost  Duties.     (Carr.) 24mo,  morocco, 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco, 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco, 

Durand's  Resistance  and  Propulsion  of  Ships Svo, 

*  Dyer's  Handbook  of  Light  Artillery. i2mo, 

Eissler's  Modern  High  Explosives Svo, 

*  Fiebeger's  Text-book  on  Field  Fortification Small  Svo, 

Hamilton's  The  Gunner's  Catechism iSmo, 

*  Hoff's  Elementary  Naval  Tactics Svo, 

Ingalls's  Handbook  of  Problems  in  Direct  Fire Svo, 

*  Ballistic  Tables Svo. 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II .  .  Svo,  each, 

*  Mahan's  Permanent  Fortifications.     (Mercur.) Svo,  half  morocco. 

Manual  for  Courts-martial i6mo   morocco, 

*  Mercur's  Attack  of  Fortified  Places i2mo, 

*  Elements  of  the  Art  of  War Svo, 

Metcalf 'sXost  of  Manufactures — And  the  Administration  of  Workshops,  Public 

and  Private Svo, 

*  Ordnance  and  Gunnery i2mo, 

Murray's  Infantry  Drill  Regulations i8mo,  paper, 

*  Phelps's  Practical  Marine  Surveying Svo, 

Powell's  Army  Officer's  Examiner i2mo, 

Sharpe's  Art  of  Subsisting  Armies  in  War iSmo,  morocco, 

a 


7 

00 

7 

50 

2 

00 

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25 

15 

00 

5 

00 

3 

00 

4 

00 

2 

00 

I 

00 

I 

50 

4 

00 

I 

50 

6 

00 

7 

50 

I 

50 

2 

00 

4 

00 

5 

00 

5 

00 

♦  Walke's  Lectures  on  Explosives 8vo,  4  00 

*  Wheeler's  Siege  Operations  and  Military  Mining 8vo,  2  00 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i   50 

Young's  Simple  Elements  of  Navigation i6mo.  morocco,  i  00 

Second  Edition,  Enlarged  and  Revised i6mo,  morocco  2  00 


ASSAYING. 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  00 

Miller's  Manual  of  Assaying 1 2mo,  i  00 

O'DriscoU's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  00 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  00 

Ulke's  Modern  Electrolytic  Copper  Refining Svo,  3  00 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process 1 2mo .  I  50 


ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

raig's  Azimuth 4to ,  3  50 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  00 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy Svo,  3  00 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy Svo,  3  00 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  00 


BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,  1  25 

Thome  and  Bennett's  Structural  and  Physiological  Botany i6mo,  2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) Svo,  2  00 


CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables i2mo,  t  25 

Allen's  Tables  for  Iron  Analysis Svo,  3  00 

Arnold's  Compendium  of  Chemistry.     (Mandel.)     (/n  ■preparation.) 

Austen's  Notes  for  Chemical  Students i2mo,  i  50 

Bernadou's  Smokeless  Powder. — Nitro-ceUulose,  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Bolton's  Quantitative  Analysis Svo,  i  50 

*  Browning's  Introduction  to  the  Rarer  Elements Svo,  i  50 

Brush  and  Penfield's  Manual  of  Determinative  Mineralogy Svo,  4  00 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.  (Boltwood.)  . . .  .Svo  3  00 

Cohn's  Indicators  and  Test-papers i2mo,  2  00 

Tests  and  Reagents Svo,  3  00 

Copeland's  Manual  of  Bacteriology.     {In  preparation.) 

Craft's  Short  Course  in  Qualitative  Chemical  Analysis.  (Schaeffer.). . .  .  i2mo,  2  00 

Drechsel's  Chemical  Reactions.     (Merrill.) i2mo,  i   25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.)     (Shortly.) 

Eissler's  Modern  High  Explosives Svo,  4  00 

3 


Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  00 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) i2mo,  i   25 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i    50 

Fowler's  Sewage  Works  Analyses 12 mo,  2  00 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,  5  00 

Manual  of  QuaUtative  Chemical  Analysis.     Parti.    Descriptive.     (Wells.) 

»                                           8vo,  3  00 

System   of   Instruction    in    Quantitative    Chemical   Analysis.      (Cohn.) 
2  vols.    (Shortly.) 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     ( Woll.) i2mo.  2  00 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (MandeL) 8vo,  4  00 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo.  i  50 

Hinds's  Inorganic  Chemistry 8vo,  3  00 

*  Laboratory  Manual  for  Students i2mo,  75 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,  2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) 8vo,  2  50 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  00 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,  r  00 

Keep's  Cast  Iron 8vo,  2  50 

Ladd's  Manual  of  Quantitative  Chemical  Analysis 12  mo  i   00 

Landauer's  Spectrum  Analysis.    (Tingle.) 8vo,  3  00 

Lassar-Cohn's  Practical  Urinary  Analysis.     (Lorenz.) i2mo,  i  00 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control.     (In  preparation.) 

LiJb's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz.)  i2mo,  i  00 

Mandel's  Handbook  for  Bio-chemical  Laboratory i2mo,  1   50 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  4  00 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i   25 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .  i2mo,  1  00 

Miller's  Manual  of  Assaying i2mo,  i  00 

Mixter's  Elementary  Text-book  of  Chemistry 1 2mo ,  i   50 

Morgan's  Outline  of  Theory  of  Solution  and  its  Results 12 mo,  i   00 

Elements  of  Physical  Chemistry i2mo.  2  00 

Nichols's  Water-supply.     (Considered  mainly  from  a  Chemical  and  Sanitary 

Standpoint,  1883.) 8vo,  2  50 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,  2  00 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores Svo,  2  00 

Ost  and  Kolbeck's  Text-book  of  Chemical  Technology.     (Lorenz — Bozart.) 

(In  preparation.) 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 
Pictet's   The    Alkaloids   and    their   Chemical   Constitution.      (Biddle.)      (In 
preparation . ) 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo,  i  50 

Poole's  Calorific  Power  of  Fuels Svo,  3  00 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  00 

Richardsand  Woodman's  Air  ,Water,  and  Food  from  a  Sanitary  Stand  point.  Svo,  2  00 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  1  00 

Cost  of  Food,  a  Study  in  Dietaries i2mo,  i  00 

*  Richards  and  WilUams's  The  Dietary  Computer Svo,  i  50 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. — 

Non-metallic   Elements.) Svo,  morocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying Svo,  3  00 

4 


d  eal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  00 

Schimpf's  Text-book  of  Volumetric  Analysis izmo,  2  50 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6pio,  morocco,  3  00 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .i6mo,  morocco,  2  00 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

•  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

•  Descriptive  General  Chemistry 8vo  3  00 

Treadwell's  Qualitative  Analysis.     (Hall.) 8vo,  3  00 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

Wells's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students i2mo,  i   50 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wiechmann's  Sugar  Analysis Small  8vo.  2  So 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process 1 2mo  i   50 

Wulling's  Elementary  Course  in  Inorganic  Pharmaceutical  and  Medical  Chem- 
istry  1 2mo ,  2  00 

CIVIL   ENGINEERING. 

BRIDGES  AND    ROOFS.       HYDRAULICS.       MATERIALS    OF   ENGINEERING. 
RAILWAY   ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments i2mp,    3  00 

Bixby's  Graphical  Computing  Table Paper,  19*  X  24}  inches  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  CanaL     (Postage 

27  cents  additional.) 8vo,  n*.     3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,    2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,    i  00 

EUiott's  Engineering  for  Land  Drainage i2mo,    i  50 

Practical  Farm  Drainage i2mo,    i  00 

Folwell's  Sewerage.     (Designing  and  Maintenance.) Svo, 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten Svo, 

French  and  Ives's  Stereotomy Svo, 

Goodhue's  Municipal  Improvements i2mo, 

Goodrich's  Economic  Disposal  of  Towns'  Refuse Svo, 

Gore's  Elements  of  Geodesy Svo, 

Hayford's  Text-book  of  Geodetic  Astronomy Svo, 

Howe's  Retaining  Walls  for  Earth i2mo, 

Johnson's  Theory  and  Practice  of  Surveying Small  Svo, 

Statics  by  Algebraic  and  Graphic  Methods Svo, 

Kiersted's  Sewage  Disposal i2mo, 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.)  i2mo, 
Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo 

*  Descriptive  Geometry 8vo, 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy Svo, 

Elements  of  Sanitary  Engineering Svo, 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco, 

Nugent's  Plane  Surveying 8 vo , 

Ogden's  Sewer  Design i2mo, 

Patton's  Treatise  on  Civil  Engineering Svo,  half  leather. 

Reed's  Topographical  Drawing  and  Sketching 4to, 

Rideal'slSewage  and  the  Bacterial  Purification  of  Sewage Svo, 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry Svo, 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) Svo, 

5 


3 

00 

3 

SO 

2 

50 

I 

75 

3 

50 

2 

50 

3 

00 

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25 

4 

00 

2 

00 

I 

25 

2 

00 

5 

00 

I 

50 

2 

50 

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00 

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00 

3 

50 

2 

00 

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50 

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00 

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50 

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5 

00 

6 

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6 

50 

5- 

roc 

5 

50 

3 

oo 

2 

50 

I 

25 

4 

oo 

3" 

'so 

Sondericker's  Graphic   Statics,   witn  i.pplications   to   Trusses,  Beams,  and 
Arches.     (Shortly.) 

*  TraiitwLne's  Civil  Engineer's  Pocket-book i6mo,  morocco, 

Wait's  Engineering  and  Architectural  Jixrisprudence 8vo, 

Sheep, 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo, 

Sheep, 

Law  of  Contracts 8vo, 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo, 

Webb's  Problems  in  the  U?e  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco, 

♦  Wheeler's  Elementary  Course  of  Civil  Engineering 8vo, 

Wilson's  Topographic  Surveying 8vo, 


BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  oo 

*         Thames  River  Bridge 4to,  paper,  s  oo 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  50 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  00 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  00 

Fowler's  Coffer-dam  Process  for  Piers 8vo,  2  50 

Greene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Howe's  Treatise  on  Arches 8vo  4  00 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel 8vo,  2  00 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern   Framed   Structures Small  4to,  10  00 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I. — Stresses  in  Simple  Trusses 8vo,  2  50 

Part  II. — Graphic  Statics 8vo,  2  50 

Part  III. — Bridge  Design.     4th  Edition,  Rewritten 8vo,  2  50 

Part  IV. — Higher  Structures 8vo,  2  50 

Morison's  Memphis  Bridge 4to,  10  00 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers. . .  i6mo,  morocco,  3  00 

Specifications  for  Steel  Bridges i2mo,  i   25 

Wood's  Treatise  on  the  Theory  of  the  Construction  of  Bridges  and  Roofs.8vo,  2  00 
Wright's  Designing  of  Draw-spans: 

Part  I.  — Plate-girder  Draws 8vo,  2  50 

Part  II. — Riveted-truss  and  Pin-connected  Long-span  Draws 8vo,  2  50 

Two  parts  in  one  volume 8vo,  3  50 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from  an 

Orifice.     (Trautwine. ) 8vo,  2  00 

Bovey's  Treatise  on  Hydraulics 8vo,  5  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

CoflSn's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Folwell's  Water-supply  Engineering 8vo,  4  00 

Frizell's  Water-power 8vo,  5  00 


Fuertes's  Water  and  Public  Health i zmo,  i  50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) 8vo,  4  00 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  00 

Hazlehurst's  Towers  and  Tanks  for  Water- works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  00 

Mason's    Water-supply.     (Considered    Principally    from    a    Sanitary    Stand- 
point.)    3d  Edition,  Rewritten 8vo,  4  00 

Merriman's  Treatise  on  Hydraulics,     gth  Edition,  Rewritten 8vo,  5  00 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  00 

Schuyler's   Reservoirs  for  Irrigation,   Water-power,  and   Domestic   Water- 
supply  Large  8vo,  5  00 

**  Thomas  and  Watt's  Improvement  of  Riyers.     (Post.,  44  c.  additional),  4to,  6  00 

Turneaure  and  Russell's  Public  Water-supplies 8vo.  5  00 

Wegmann's  Desien  and  Construction  of  Dams 4to,  5  00 

Water-suoolv  of  the  City  of  New  York  from  1658  to  189S 4to,  10  00 

Weisbach's  Hydraulics  and  Hydraulic  Motors.     (Du  Bois.) 8vo,  5  00 

Wilson's  Manual  of  Irrigation  Engineering Small  8vo,  4  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo, '  3  00 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  00 


MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo, 

Roads  and  Pavements 8vo, 

Black's  United  States  Public  Works Oblong  4to, 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo, 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edi- 
tion, Rewritten 8vo, 

Byrne's  Highway  Construction 8vo, 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo. 

Church's  Mechanics  of  Engineering 8vo, 

Du  Bois's  Mechanics  of  Engineering.     VoL  I Small  4to, 

Johnson's  Materials  of  Construction Large  8vo, 

Keep's  Cast  Iron 8vo, 

Lanza's  Applied  Mechanics 8vo, 

Martens's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo, 

Merrill's  Stones  for  Building  and  Decoration 8vo, 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,    4  00 

Strength  of  Materials lamo,    i  00 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,    2  00 

Patton's  Practical  Treatise  on  Foundations 8vo,    5  00 

Rockwell's  Roads  and  Pavements  in  France i2mo,    i  25 

Smith's  Wire :  Its  Use  and  Manufacture Small  4to,    3  00 

Materials  of  Machines 1 2mo,    i  00 

Snow's  Principal  Species  of  Wood 8vo,    3  50 

Spalding's  Hydraulic  Cement i2mo,    2  00 

Text-book  on  Roads  and  Pavements i2mo,    2  00 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,    8  00 

Part  I. — Non-metaUic  Materials  of  Engineering  and  Metallurgy 8vo,    2  00 

Part  II. — Iron  and  Steel 8vo,    3  50 

Part  III. — A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,    2^0 

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Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  00 

TiUson's  Street  Pavements  and  Paving  Materials 8vo,  4  00 

Waddell's  De  Pontibus.     (A  Pocket-book  for  Bridge  Engineers.).  .  i6mo,  mor.,  3  00 

Specifications  for  Steel  Bridges , i2mo,  1  25 

"Wood's  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on  the  Pres- 
ervation of  Timber 8vo,  2  00 

Elements  of  Analytical  Mechanics 8vo,  3  00 


RAILWAY  ENGINEERING. 

Andrews's  Handbook  for  Street  Railway  Engineers.     3X5  inches,  morocco,  i   25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Brooks's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  1  50 

Butts's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  1  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.    i6mo,  morocco,  4  00 

Dredge's  History  of  the  Pennsylvania  Railroad:    (1879) Paper,  5  00 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills,  4to,  half  mor.,    25  00 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide i6mo,  mor.,  2   50 

Howard's  Transition  Curve  Field-book i6mo   morocco  i  so 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments    8vo,  I   00 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  00 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco.  .^  00 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  00 

Pratt  and  Alden's  Street-railway  Road-bed Svo,  2  00 

Searles's  Field  Engineering i6mo,  morocco,  3  00 

Railroad  Spiral i6mo,  morocco  i   50 

Taylor's  Prismoidal  Formulae  and  Earthwork Svo,  1   50 

*  Trautwine's  Method  of  Calculating  the  Cubic  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams Svo,  2  00 

he  Field  Practice  of  .Laying    Out    Circular    Curves    for    Raibroads. 

i2mo,  morocco,  2  50 

*  Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction.     2d  Edition,  Rewritten i6mn.  morocco,  5  00 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  Svo,  5  00 


DRAWING. 

Barr's  Kinematics  of  Machinery Svo,  2  50 

•  Bartlett's  Mechanical  Drawing Svo,  3  00 

Coolidge's  Manual  of  Drawing Svo,  paper,  i  00 

Durley's  Kinematics  of  Machines Svo,  4  00 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective Svo,  2  00 

Jones's  Machine  Design: 

Part  I. — Kinematics  of  Machinery Svo, 

Part  II. — Form,  Strength,  and  Proportions  of  Parts Svo, 

MacCord's  Elements  of  Descriptive  Geometry Svo, 

Kinematics;   or.  Practical  Mechanism Svo, 

Mechanical  Drawing 4to, 

Velocity  Diagrams ." Svo, 

♦  Mahan's  Descriptive  Geometry  and  Stone-cutting Svo, 

Industrial  Drawing.    (Thompson.) Svo, 

Reed's  Topographical  Drawing  and  Sketching 4to, 

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Reid's  Cotirse  in  Mechanical  Drawing 8vo, 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design.  .8vo, 

Robinson's  Principles  of  Mechanism 8vo, 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) Svo, 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  .  i2mo, 

Drafting  Instruments  and  Operations i2mo, 

Manual  of  Elementary  Projection  Drawing i2mo. 

Manual  of  Elementary  Broblems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo, 

Plane  Problems  in  Elementary  Geometry i2mo, 

Primary  Geometry 1 2mo, 

Elements  of  Descriptive  Geometry,  Shadows,  and^Perspective 8vo, 

General  Problems  of  Shades  and  Shadows Svo, 

Elements  of  Machine  Construction  and  Drawing Svo, 

Problems.  Theorems,  and  Examples  in  Descriptive  Geometry Svo, 

Weisbach's  Kinematics  and  the  Power  of  Transmission.       v Hermann  an'* 

Klein.)   Svo,     5  00 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2mo,    2  00 

Wilson's  Topographic  Surveying Svo,    3  50 

Free-hand  Perspective Svo,    2  50 

Free-hand  Lettering.     {In  preparation.) 
Woolf's  Elementary  Course  in  Descriptive  Geometry Large  Svo,    3  00 


'ELECTRICITY  AND   PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.).  ......  .Small  Svo.  3  00 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements i2mo,  1  00 

Benjamin'slHistory  of  Electricity Svo,  3  00 

Voltaic  CelL 8vo,  3  00 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Boltwood.).  .Svo,  3  00 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo,  3  00 

Dawson's  "Encineering"  and  Electric  Traction  Pocket-book. .  lomo,  morocco,  4  00 

Klather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Gilbert's  De  Magnete.     (Mottelay.) Svo,  2  50 

Hohnan's  Precision  of  Measurements Svo,  2  00 

Telescopic  Mirror-scale  Method,  Adjustments,  and  Tests Large  »vo  75 

Lanaauer's  Spectrum  Analysis.    (Tingle.) Svo,  3  00 

Le  ChateUer's  High-temperature  Measurements.  (Boudouard — iJurgess.  )i2mo,  3   00 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz.)  i2mo,  i   00 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.     Vols.  I.  and  11.  Svo,  each,  6  00 

*  Michie.     Elements  of  Wave  Motion  Relating  to  Sound'and  Light. ...  ^.  .Svo,  4  00 
Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (FishoacK.  i i2mo,  2  50 

*  Parshall  and  Hobart's  Electric  Generators Small  4to.  half  morocco,  10  00 

*  Rosenberg's  Electrical  Engineering.    (HaldaneGee — Kinzbninner.).  . .  .Svo,  i   50 
Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     (In  preparalioi'.' 

Thurston's  Stationary  Steam-engines Svo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat Svo,  1   50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  Svo,  2  00 

Ulke's  Modern  Electrolytic  Copper  Refining Svo,  3  00 


LAW. 

*^Dayis's  Elements  of  Law Svo,  2  50 

•  Treatise  on  the  MiUtary  Law  of  United  States Svo,  7  00 

♦  Sheep,  7  50 
Manual  for  Courts-martial i6mo,  morocco,  1   50 


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6 

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3 

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2 

50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo, 

Sheep, 
Law  of  Operations  Preliminary  to  Construction  in  Engineering'and  Archi- 
tecture      8vo, 

Sheep, 

Law  of  Contracts 8vo, 

Winthrop's  Abridgment  of  Military  Law lamo. 


MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,    2  50 

Bolland's  Iron  Founder i2mo,    2  50 

"  The  Iron  Founder,"  Supplement i2mo,    2  50 

Encyclopedia  of  Founding  and  Dictionary  oflFoundry  Terms  Used  in  the 

Practice  of  Moulding 1 2mo,    3  00 

Eissler's  Modem  High  Explosives 8vo,    4  00 

Efifront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,    3  00 

Fitzgerald's  Boston  Machinist i8mo,    i  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,    i  00 

Hopkins's  Oil-chemists'  Handbook 8vo,    3  00 

Keep's  Cast  Iron 8vo,    2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  SpeciaI]Reference  to  State 

Control.     (In  preparation.) 

Metcalf 's  Steel.     A  Manual  for  Steel-users i2mo,    2  00 

Metcalfe's  Cost  of  Manufactures — And  the  Administration    of  Workshops, 

Public  and  Private 8vo, 

Meyer's  Modern  Locomotive  Construction 4to, 

*  Reisig's  Guide  to  Piece-dyeing 8vo, 

Smith's  Press-working  of  Metals 8vo, 

Wire:   Its  Use  and  Manufacture Small  4to, 

Spalding's  Hydraulic  Cement i2mo, 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco, 

Handboo'K  tor  bugar  Manufacturers  ana  their  Chemists.. .  i6mo,  morocco, 
Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion   8vo, 

*  Walke's  Lectures  on  Explosives 8vo, 

West's  American  Foundry  Practice i2mo. 

Moulder's  Text-book < i2mo, 

Wiechmann's  Sugar  Analysis Small  8vo, 

Wolff's  Windtnill  as  a  Prime  Mover 8vo, 

Woodbury's  Fire  Protection  of  Mills 8vo, 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  i  50 

♦  Bass's  Elements  of  Differential  Calculus z2mo,  4  00 

Briggs's  Elements  of  Plane  Analytic  Geometry i2mo,  i  00 

Chapman's  Elementary  Course  in  Theory  of  Equations i2mo,  i  50 

Compton's  Manual  of  Logarithmic  Computations lamo,  i  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  i  50 

♦  Dickson's  College  Algebra Large  i2mo,  i  50 

♦  Introduction  to  the  Theory  of  Algebraic  Equations   Largeliamo,  i  25 

Halsted's  Elements  of  Geometry Svo,  i  75 

Elementary  Synthetic  Geometry Svo,  1  50 

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♦Johnson's  Three-place  Logarithmic  Tables:    Vest-pocket  size paper,  is 

100  copies  for  5  00 

•  ■'                                      Mounted  on  heavy  cardboard,  8  X  lo  inches,  as 

10  copies  for  2  00 

Elementary  Treatise  on  the  Integral  Calculus Small  8vo,  i  50 

Curve  Tracing  in  Cartesian  Co-ordinates i2mo,  i  00 

Treatise  on  Ordinary  and  Partial  Differential  Equations Small  8vo,  3  50 

Theory  of  Errors  and  the  Method  of  Least  Squares i2mo,  i  50 

•  Theoretical  Mechanics 1 2mo,  3  00 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.)  i2mo,  2  00 

•  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,  3  00 

Trigonometry  and  Tables  published  separately Each,  2  00 

Maurer's  Technical  Mechanics.     (In  preparation.) 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  00 

Merriman's  Method  of  Least  Squares 8vo,  a  00 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. Sm.,  8vo,  3  00 

Differential  and  Integral  Calculus.     2  vols,  in  one Gmall  8vo,  2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  00 

Trigonometry:  Analytical,  Plane,  and  Spherical i2mo,  i  00 

MECHANICAL   ENGINEERING. 
MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Baldwin's  Steam  Heating  for  Buildings i2mo, 

Barr's  Kinematics  of  Machinery 8vo, 

•  Bartlett's  Mechanical  Drawing 8vo, 

Benjamin's  Wrinkles  and  Recipes izmo. 

Carpenter's  Experimental  Engineering 8vo, 

Heating  and  Ventilating  Buildings 8vo, 

Clerk's  Gas  and  Oil  Engine Small  8vo, 

Coolidge's  Manual  of  Drawing 8vo,    paper, 

Cromwell's  Treatise  on  Toothed  Gearing i2mo. 

Treatise  on  Belts  and  Pulleys i2mo, 

Durley's  Elinematics  of  Machines 8vo, 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo. 

Rope  Driving i2mo. 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo, 

HaU's  Car  Lubrication i2mo, 

Button's  The  Gas  Engine.     (In  preparation.) 
Jones's  Machine  Design: 

Part   I. — Kinematics  of  Machinery v-  -Svo,    i  50 

Part  II. — Form,  Strength,  and  Proportions  of  Parts. 8vo, 

Kent's  Mechanical  Engineer's  Pocket-book i6mo,    morocco, 

Kerr's  Power  and  Power  Transmission 8vo, 

MacCord's  Kinematics;  or,  Practical  Mechanism 8vo, 

Mechanical  Drawing 4to, 

Velocity  Diagrams 8vo, 

Mahan's  Industrial  Drawing.    (Thompson.) 8vo, 

Poole's  Calorific  Power  of  Fuels 8vo, 

Reid's  Course  in  Mechanical  Drawing 8vo.    2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design.  .8vo,    3  00 

Richards's  Compressed  Air i2mo,    1  50 

Robinson's  Principles  of  Mechanism 8vo,    3  00 

Smith's  Press-working  of  Metals -  -  8vo     3  00 

Thurston's  Treatise  on    Friction  and    Lost  Work   in    Machinery   and   Mil 

Work 8vo,    3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  1 2mo,    i  00 

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Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  the  Power  of  Transmission.      Herrmann — 

Klein.) 8vo,  5  00 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  00 

Hydraulics  and  Hydraulic  Motors.     (Du  Bois.) 8vo,  5  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Turbines 8vo,  2  50 

MATERIALS  OF  ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition, 

Reset 8vo ,  750 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Johnson's  Materials  of  Construction Large  Svo,  6  00 

Keep's  Cast  Iron Svo  2  50 

Lanza's  Applied  Mechanics Svo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) Svo,  7  50 

Merriman's  Text-book  on  the  Mechanics  of  Materials Svo,  4  00 

Strength  of  Materials i2mo,  i  00 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo  2  00 

Smith's  Wire:   Its  Use  and  Manufacture Small  4to,  3  00 

Materials  of  Machines i2mo,  i  00 

Thurston's  Materials  of  Engineering 3  vols  ,  Svo,  8  00 

Part    II. — Iron  and  Steel Svo,  3  50 

Part  III. — A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo,  2  50 

Text-book  of  the  Materials  of  Construction Svo  5  00 

Wood's  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on  the 

Preservation  of  Timber Svo,  2  00 

Elements  of  Analytical  Mechanics Svo,  3  00 

STEAM-ENGINES  AND  BOILERS. 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,  i   50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  T6mo,  mor.,  4  00 

Ford's  Boiler  Making  for  Boiler  Makers iSmo,  i  00 

Goss's  Locomotive  Sparks Svo,  2  00 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  00 

Button's  Mechanical  Engineering  of  Power  Plants Svo,  5  00 

Heat  and  Heat-engines Svo,  5  00 

Kent's  Steam-bo'ler  Economy Svo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector Svo.  i  50 

MacCord's  Slide-valves Svo,  2  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo,  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors Svo,  1  00 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines Svo,  5  00 

Valve-gears  for  Steam-engines Svo,  2  50 

Peabody  and  Miller's  Steam-boilers Svo,  4  00 

Pray's  Twenty  Years  with  the  Indicator Large  Svo,  2  50 

Pupln's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  25 

Reagan's  Locomotives  :  Simple,  Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) Svo,  5  00 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,  2  00 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

Snow's  Steam-boiler  Practice Svo,  3  00 

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4 

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Spangler's  Valve-gears 8vo,    2  50 

Notes  on  Thermodynamics i2mo,    i  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo, 

Thurston's  Handy  Tables 8vo. 

Manual  of  the  Steam-engine 2  vols.  8vo, 

Part  I. — History,  Structuce,  and  Theory 8vo, 

Part  n. — Design,  Construction,  and  Operation 8vo, 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo, 

Stationary  Steam-engines 8vo, 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo, 

Manual  of  Steam-boiler?  ,  Their  Designs,  Construction,  and  Operation. 8vo, 

Weisbach's  Heat,  Steam,  a    J  Steam-engines.     (Du  Bois.) 8vo, 

Whitham's  Steam-engine  1  esign 8vo, 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo. 

Wood's  Thermodynamics.  Heat  Motors,  and  Refrigerating  Machines.  .  .  .8vo, 


MECHANICS     A.ND    MACHINERY. 

Barr's  Kinematics  ot  machinery 8vo,    2  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,    7  50 

Chase's  The  Art  of  Pattern-making i2mo,    2  50 

Chordal. — Extracts  from  Letters i2mo,    2  00 

Church's  Mechanics  of  Engineering 8vo     6  00 

Notes  and  Examples  in  Mechanics 8vo,    2  00 

Compton's  First  Lessons  in  Metal-working i2mo,    i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,    i  so 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,    i  50 

Treatise  on  Belts  and  Pulleys i2mo,    1  50 

Dana's  Text-book  of   Elementary  Mechanics  for  the   Use   of   Colleges  and 

Schools i2mo,    1  50 

Dingey's  Machinery  Pattern  Making i2mo,    2  00 

Dredge's   Record   of   the   Transportation   Exhibits  Building  of   the   World's 

Columbian  Exposition  of  i8g3 4to,  half  morocco,    5  00 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.     I. — Kinematics 8vo, 

Vol.    II.— Statics 8vo, 

Vol.  III.— Kinetics 8vo, 

Mechanics  of  Engineering.     Vol.  I Small  4to, 

Vol.  II SmaU  4to, 

Durley's  Kinematics  of  Machines 8vo, 

Fitzgerald's  Boston  Machinist i6mo,    i  00 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,    3  00 

Rope  Driving i2mo,    2  00 

Goss's  Locomotive  Sparks 8vo,    2  00 

Hall's  Car  Lubrication i2mo,    i  00 

Holly's  Art  of  Saw  Filing i8mo  75 

♦  Johnson's  Theoretical  Mechanics i2mo,  .  3  00 

Statics  by  Graphic  and  Algebraic  Methods 8vo,    2  00 

Jones's  Machine  Design: 

Part   I. — Kinematics  of  Machinery 8vo,    i   50 

Part  II.^Form,  Strength,  and  Proportions  of  Parts 8vo,    3  00 

Kerr's  Power  and  Power  Transmission 8vo,    2  00 

Lanza's  Applied  Mechanics 8vo,    7  50 

MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,    5  00 

Velocity  Diagrams 8vo,    i   50 

Maurer's  Technical  Mechanics.     (In  preparation^) 

13 


3 

50 

4 

00 

3 

50 

7 

50 

10 

00 

4 

00 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  00 

•^Michie's  Elements  of  Analytical  Mechanics 8vo,  4  00 

Reagan's  Locomotives:  Simple,  Compound,  and  Electric 1 2mo,  2  50 

Reid's  Course^in  Mechanical  Drawing 8vo,  2  00 

Text -book  of^Mechanical  Drawing  and  Elementary  Machine  Design.  .8vo,  3  00 

Richards's  Compressed  Air 1 2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     (In  preparation.) 

Sinclair's  Locomotive-engine  Running  and]Management izmo,  2  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

<      Materials  of  Machines i2mo,  1  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  00 

Thurston's  Treatise  on  Friction  and   Lost  Work   in  Machinery  and   Mill 

"Work 8vo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  i2mo,  i  00 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics  land    the   Power  of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  00 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein. ).8vo,  5  00 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  00 

Principles  of  Elementary  Mechanics i2mo,  i  25 

Turbines 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

METALLURGY. 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

VoL    I. — Silver 8vo,  7  50 

VoL    n. — Gold  and  Mercury 8vo,  7  50 

**  Iles's  Lead-smelting.     (Postage  g  cents  additional.) i2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.   (Boudouard — Burgess.) .  i2mo,  3  00 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  00 

Smith's  Materials  of  Machines i2mo,  i  00 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  00 

Part   II. — Iron  and  Steel 8vo,  3  50 

Part  III.— A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and   their 

Constituents 8vo,  2  50 

Hike's  Modern  Electrolytic  Copper  Refining 8vo,  3  00 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.     Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  00 

Map  of  Southwest  Virginia Pocket-book  form,  2  00 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  00 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  00 

Cloth,  1  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "System  of  Mineralogy.". . .  .Large  8vo,  i  00 

Text-book  of  Mineralogy 8vo,  4  00 

Minerals  and  How  to  Study  Them . . .  : i2mo,  1   50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  i  00 

Manual  of  Mineralog^y  and  Petrography i2mo,  2  00 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.     (Smith.)  Small  8vo,  2  00 

14 


♦  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  o  50 
Rosenbusch's   Microscopical   Physiography   of   the    Rock-making   Minerals. 

(Iddings.) 8vo,  5  00 

•  Tillman's  Text-book  of  Important  Minerals  and  Docks 8vo,  2  00 

Williams's  Manual  of  Lithology 8vo,  3  00 

MINING. 

Beard's  Ventilation  of  Mines i2mo,  2  so 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  00 

Map  of  Southwest  Virginia Pocket-book  form,  2  00 

♦  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills. 

4to,  half  morocco,  25  00 

Eissler's  Modem  High  Explosives 8vo,  4  00 

Fowler's  Sewage  Works  Analyses i2mo,  2  00 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo,  2  50 

Ihlseng's  Manual  of  Mining .    8vo,  4  00 

**  Iles's  Lead-smelting.     (Postage  gc.  additional.) i2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  00 

•  Walke's  Lectures  on  Explosives 8vo,  4  00 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  50 

Hydraulic  and  Placer  Mining i2mo,  2  00 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo,  i  25 

SANITARY  SCIENCE. 

Copeland's  Manual  of  Bacteriology.     {In  preparation.) 

Folwell's  Sewerage.     (Designing,  Construction   and  Maintenance.; 8vo,  3  00 

Water-supply  Engineering 8vo,  4  00 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration    Works i2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  1  00 

Goodrich's  Economical  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  00 

Kiersted's  Sewage  Disposal i2mo,  r  25 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

ControL     (In  preparation.) 
Mason's    Water-supply.     (Considered    Principally   from    a    Sanitary   Stand- 
point.)    3d  Edition,  Rewritten 8vo,  4  00 

Examination  of  Water.     (Chemical  and  BacteriologicaL) 12 mo,  i  25 

Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  00 

Nichols's  Water-supply.     (Considered  Mainly  from  a  Chemical  and  Sanitary 

Standpoint.)     (1883.) 8vo,  2  50 

Ogden's  Sewer  Design i2mo,  2  00 

*  Price's  Handbook  on  Sanitation i2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  00 

Cost  of  Living  as  Modified  by  Sanitary^Science i2mo,  i  00 

xUchards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point   8vo,  3  00 

*  Richards  and  Williams's  The  DietarylComputer 8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Woodhull's  Notes^and  Military  Hygiene i6mo,  i   50 

15 


MISCELLANEOUS. 

Barker's  Deep-sea  Soundings 8vo,  2  00 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds. .      8vo,  4  00 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition.'Digestibility,  and  Nutritive  Value  of  Food.    Mounted  chart,  i   25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.  Small  8vo,  3  00 

Rotherham's  Kmphasized  New  Testament Large  8vo,  2  00 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance, 
and  Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small 

Hospital i2mo,  i   25 

HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Grammar  of  the  Hebrew  Language 8vo,  3  00 

Elementary  Hebrew  Grammar i2mo,  i   25 

Hebrew  Chrestomathy 8vo ,  2  00 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  00 

Letteris's  Hebrew  Bible 8vo,  2  25 

16 


University  of  California 

SOUTHERN  REGIONAL  LIBRARY  FACILITY 

Return  this  material  to  the  library 

from  which  it  was  borrowed. 


AUPL 

JUN  101989 

QUAPiTR  LOAri 

JUN  15 1989 

REC'D  AUPL. 

NOV  24  1989 

REC'D  AUPL. 

RECEIVED 

APR  0  8  1993 
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